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- 1. CHAPTER 3 FUNCTIONS MAPPING Consider the function f defined by f : x + 2x+ 1. If we input the value 2 for x, f gives 5 as output and we say that this function maps 2 to 5, or 2 + 5 . Similarly f maps -2 to -3, -1 to -1, 0 to 1, . . . . This mapping can be illustrated as follows. In this example, one value of the independent variable x maps to just one unique value of the dependent variable f(x). Such a function is said to define a one-one mapping. Now consider the function f given by f : x * xL-b + 6. Thisfunctionmaps -1 to 9,0 to 6, I to 5, 2 to 6,3 to 9,.... i.e. x+f'-2x*6 6 9 43 f(x)
- 2. M Mathematics - The Core Gourse for A-!eve! In this exztmple one value of x again gives rise to just one value of f(x) but this value of /(x) is not necessarily uniquely obtained. For example, both 0 and 2 map to 6. Thus there are at least some values of f(x) which arise from more than one value of x. Such a function is called a many-one mapping. Hence we may regard a function as a 'rule' for mapping a number o to a number b. However the strict mathematical definition of the word function is restricted to those relationships for which one input value gives rise to iust one output value. So the two relationships considered aboye,viz. x + 2x + I and 19 + x2 -2x * 6 are functions under this definition. But if we consider the relationship x + tt/x, w0 find that if we input the value 4, say, for x we get as output the two values 2 and -2. Thus x + tlx is not a function, although it does define a perfectly good mapping for positive values of x. GRAPHICAL REPRESENTATION OF FUNCTIONS (CURVE SKETCHING} Consider again the function f(x) = x2 -2x + 6. For any arbitrarily chosen value of x, the corresponding value of /(x) can be calculated, e.g. Having arranged the values of f(x) in the order of ascending values of x, we can see that there is some pattern to the values obtained for f(x). This pattern is easier to visualise if the results are displayed graphically, i.e. if the values of /(x) on a vertical number line are plotted against the corresponding values of x on a horizontal number line. The following graph has been drawn using only eight values of x, separated by intervals of one unit. We could choose values of x from a larger range and use smaller intervals. The corresponding values of /(x) so obtained would all be found to lie on the smooth curve we have already drawn. Therefore this curve is a graphical representation of the varying values of /(x) obtained from varying values of x and all possible related values of /(x) and x lie on the curve.
- 3. Fulrsti(xts tt We can now use the graph to deduce some properties of the function. (u) The lowest point on the curve is where x : I and f(x) - 5 so although x, as the independent variable, can be given any real value, the corresponding value of/(x) is always greater than 5, or equal to 5 when x : I . We say that the function has a least value of 5. (b) The curve is symmetrical about the line x : l, as two values of x equi- distant from x: l, (1 + a and l-a), 8tve the same value for f(x). (c) Conversely, for any one value of /(x), there are two corresponding values for x of theform l*a and l-a. All these properties can be confirmed algebraically as follows: completing the square on R.H.S gives f(x) - x2 - bc + 6 - (x - l)' + 5 (a) Mhatever value we substitute for x, (x - l)' is always positive as it is a squared quantity. Therefore the least value of (, - 1)' is zero and this occurs when x : I . Hence the lowest value that f(x) can have is 5, i.e. f(x) has a least value of 5 whery x : I . (b) Consider two values of x symmetrically placed on either side of x : l, i.e. x:l*a and x:l-a f(l+a)- a2+5
- 4. 46 Mathematics - The Core Course for AJevel Therefore values of x that are symmetrical about x : I give the same value for /(x). (c) Conversely, for any given values of f(x), c say, the corresponding values of x are given by the solution of the equation c : x2-2J+6 or x2-2x*6-c : O From the formula, the roots of this equation are x: l+/c-5 and x:1-Jc-5 Thusfor x tohaverealvalues, c>5, i.e.f(x) hasaleastvalueof 5 and these two realvalues of .x, corresponding to one value of f(x), are symmetricalabout x: l. DOMAIN AND BANGE For the function f : x -+ x2 -2x + 6 we have assumed that we can input any real value of x. However for a full definition of a function, the set of permissible input values must be stated. So the function illustrated above is fully defined as f:x*x'-2x+6, x€lR (in this context x € IR means that x can be any member of the set of real numbers). The set of input values for which any function is defined is called the domain of the function, and the domain is not always the complete set of real numbers as we shall see in the following examples. The analysis of the function f(x) = x2 -2x + 6 also showed that, for the domain x € IR, the output values were limited, i.e. f(x) > 5. Once the domain of any function is defined there is a corresponding set of output values. This set of output values is called the range of the function (or the image-set). Hencetherangeof / where 7'.x->x2-2x*6, x€lR is /(x) e IR, f(x) > s EXAMPLES 3a 1) Draw a sketch graph ofthe function defined by f: x -, 2x + 1, x € IR and state its range.
- 5. Functions 47 The following arbitrary set of values for x gives from which we get the sketch graph on the right. As x can be any real number, this line is infinite in both directions so the range of / also is the complete set of real numbers. 2) For the function illustrated in Example l, A is the point on the line for which x : -1 and B is the point for which x : 2. Define the function represented by the line segment AB and state its range. The mapping represented by the line segment AB is x'+ 2x *l but only forvaluesof x from -l to 2.'' Sothefunctionrepresentedby AB is f:x-+2xtl, xeIR, -l (x(2 The range ofthis function is the set ofvalues of /(x) corresponding to -1(x(2 i.e. the range of / is -1 </(x) ( 5. f(x "f(x)
- 6. 48 Mathematics - The Core Course for A{evel 3) State the image-set of the function defined as f(x): bc+ 1, x € {0, I ,2,3,4 For this function there are just 5 input values. The corresponding output values form the image-set. So the image-set is {r,3 ,s,7 ,91 The graphical representation of this function is the set of five points shown on the right. Note that when the domain of a function is not stated it is assumed to be x€lR. EXERCISE 3a 1) Given f:x*x' for xeIR, (r) draw the graph of / and state its range, (b) if the domain is redefined as x € R*, sketch the graph that now represents f . 2) The domain of each of the following mappings is the set of real numbers. State which of these mappings are functions. (a) s+a'*5, (b) fi+r/b, (c) c * t/c 3) In Utopia, income ta:r on earnings is calculated as follows: The first 910000 is tax free, the next 910000 is taxed at 5% and the remaining income is taxed at 10%. Taking income as'input'and'tax payable as 'output', state whether these rules for calculating tax constitute a function. If they do, state the implied domain and range. ' 4) If in Question 3 {,1 isincomeand {,T is the tax payable, express the mapping I + T as algebraic formulae of the form f : f(I), stating the values of I for which they arc valid. THE QUADRATIC FUNCTION Functions of similar form usually have properties in common, so their graphs are usually similar in shape. A knowledge of the common characteristics
- 7. Funstions tl9 of such functions will allow us to draw a sketch of the graph for any one particular function without calculating a series of coresponding values of /(x) and x. The function f(x)' = x2 -2x + 6 analysed above is a quadratic function and any function whose general form is f(x)=ac2+bx+c where a, b and c are constants, is called a quadratic function. Completing the square on the R.H.S of this expression eves .f(x) = ,V'+..?f :{+*-o'l--L.,_q':t * j-oY**) Nowwhatevervalue x takes, {%r) isconstantandequalto K say *u (' * il' " o as it is a squared quantitv' Therefoie the function has the general form f(x)= K*a (zercor a*ve quantity). So if a is positive i.e, a ) O, /(x) is at least equal to K i.e. f(x) hualeastvalueof 4*-b', occuringwhen *:-! 4a-2a When a is negative,i.e. a 10, I@) cn never be gr€ater than /C i.e. /(x) hasa greatestvalue of fu-.-b' , *h"n * = -' ! k2a Now * = - ! is the value of x corresponding to the greatest or least value 2a of /(x). Takingtwovaluesof x thataresymmetricalabout -!, r.". ,=- U 2a' 2o'k' gives ,(-**o) : t(-*-/,) : o1,z +@-! i.e. input values of x that are symmetrical about * : - | give as output the same value of /(x). I
- 8. s0 Mathematics - The Core Course for A-level The diagrams below represent the two alternative graphs of a quadratic function. e sketched nting f l-l i,lor / tb, i,l nrr:l--;l or I x:r_i_ l'"/ l^-i-i_/ I lg ,lo ,lo The curve representing any particular quadratic function can now b using this information. So, for example, to sketch the curve represe f(x) = ?-x2 -7x -4 we proceed as follows: b--7, a-2 therefore ,F *) - f(z): -? i.e. f(x) has a least value of -t when * : Z. To locate the curve accurately on the axes we need one more pair o corresponding values for x and f(x). /(0) is easy to find, i.e. /(0) - - 4. 4f@) I 1z I I I I
- 9. Functions 51 Alternatively a quick sketch of a quadratic function can be obtained as follows f(x) - 2x2 -7x-4: (2x + lXx-a) The coefficient of x2 is positive, (a ) 0), so f(x) has a least value. When f(x) - 0 the corresponding values of x are roots of the quadratic equation (2x+ lXx-4)-0 i.e. x- -+ and x-4 The average of these values (il gives the value of x about which the curve is symmetrical. Note. This method is suitable only when the function factorises. EXERCISE 3b Find the greatest or least value of the following functions. l) x2-3x+5 z)zx2-4x+5 3) 3-2x-x2 4) 7 +x-x2 5) x2 -2 6) 2x-x2 Sketch the graphs of the following quadratic functions, showing clearly the greatest or least value of /(x) and the value of x at which it occurs, where f(x) is 7) x2 -2x + 5 l0) 4 - 7x - x2 Draw a quick sketch of the graph of each of the following functions, showing the axis of symmetiy clearly. 8) x2 *4x-8 1l) x2 -to 9) 2x2 -6x +3 12) 2- 5x- 3x2 ls) (2x - lXx - 3) 18) x' 13) (x - lXx - 3) t4) (x + 2)(x - 4) 16) (1 +x)(z-x) t7) x2 -e t-75 I I I I I 19) 4-x2 20) 3-7x-6x2
- 10. 52 Mathematics - The Core Course for A-level INEOUALIT!ES Consider the real numbers 5 and 2. Now 5> 2. The introduction of an extra term on both sides leaves the inequality sign unchanged e.g. i.e. 5+4>2+4 9>6 and 5-4>2-4 1 > -2 a and b arc real numbers such that a)b a*k>b+k forall realvaluesof k ak > bk for positive values of k ak 1bk for negative values of k Multiplication of both sides by a positive number also leaves the inequality sign unchanged. g.g. 5x2)2x2 i.e. 10>4 However if we multiply both sides by a negative number, the inequality is no longer true. For example, multiplying by - 2, we find that the L.H.S. becomes - 10 and the R.H.S. becomes - 4, L.H.S. < R.H.S. since - l0< - 4i.e. then and but + + + EXAMPLE 3c Find the range of values of x satisfying the inequality x-3<2x*5 Therefore the range of values of (adding 3 to both sides) (subtracting 2x from both sides) (multiplying by - l) x that satisfies the inequality is x-3<2x*5 x(2x+8 -x<8 x) 8 x)-8
- 11. Functions 53 OUADRATIC INEOUALITI ES Any inequality relationship that involves a quadratic function is called a quadratic inequality. For example , (x -Z)(2x + l) > 0 The range of values of x satisfying this inequality can be found graphically as follows: Irt f(x) = (x - 2)(2x + 1). The diagram on the right shows a sketch of f(x) from which we see that f(x) > 0 (i.e. the curve is abovg the x-axis) for values of x greater than 2 and{ess than - +. Therefore the ranges of values of x that satisfy (x-2)(2x+l)>0 are x(-* and x)2 EXERCISE 3c Find the range (or ranges) of values of x that satisfy the following inequalities. 3) x)5x-22)2x-l<x-4 s) 2{x - l) > 3(x - 1) 6) 2(x-l)<2(x+l) 7) (x-lXx-2)>0 8) (x+lXx-2)>0 e) (x-3Xx-s)<0 l0) (2x-lXx+1)<0 ll) x2-4x>s 12) 4xz < I t7) (x - 1)' > 4x2 13) 5x2 )3x +2 14) (2-xXx + 4)<0 ls) (3-zx)(x+5)>0 16) 3x)x2 +2 18) (x- lXx +2)(x$-x) Problems inyolving Ouadratic I nequalities EXAMPLES 3d I ) Find the range of values of k for which the equation For x2 - kx * (k + 3) : 0 to have real roots 1) x*2>4-x 4) 3x-5<4
- 12. l.e . 54 Mathematics - The Core Course for A-level b2 -4ac>0 (-k)'-4(k+3)>o k2 -4k-12>o f(k) : k2-4k-12 : (k-6)(k+2) From the sketch of f(k) we see that (k - 6)(k + 2)> 0 for k<-2 and k>6. 2) Find the set of values of p for which f(x): x2 *3px*p is greater than zero for all real values of x. f(x) = x2 * 3px * p is a quadratic function of x2 is positive, f(x) has a least value. So if f(x) > 0 for all x, the least value of /(x) has to be greater than zero) i.e. f(x) has a graph of the type in the sketch. Completing the square on the R.H.S. gives f(x): (,.9'*p-Y Therefore the least value of /(x) is p -T So for f(x)> 0 for all x, 9p2 p-;>o 4p-9p'>o p(4-ep)>0 Irt s(p) = p(4 - 9p). From the sketch of g(p') we see that x and, as the coefficient of p)O and i<g o<p<3 I.e,t p(4-9p)>0 for Therefore f(x)> 0 for all real ;t o< p<t. for the set of values of p given by
- 13. Functions 55 3) Find the ranges of values of x for which 2x- 1<-x2 _ 4<12 There are two inequality relationships here vtz: (u) 2x*1<x2-4 and (b) x2-4<12 and we are looking for the ranges of values of x that satisfy both of these inequalities. (r) Zx-l1xz -4 -x2 *2x+3<o xz -2x-3>o If d(x)= x2-2x-3 : (x - 3)(x + 1) the graph of f{x) is For f(x) > 0, x( I and lllustrating these ranges For s@) < 0 x)3 -4 (x <4 on a number line (b) x2- 4<12 x2 - 16 < o If s(x) - x2 - 16 :(x-4@+4) the graph of g(x) is we see that the ranges of values of lines overlap , are ,,, F,>-5 -4 -3 -2 -1 0 I 2 3 4 5 * x that satisfy both (a) and (b), i.e. where the -4(x<-1 and 3<x<-4
- 14. 56 Mathematics - The Core Course for A'level EXERCISE 3d 1) Find the range of values of x for which 3 -x < 4(x -2)' 2) Find the range of values of x for which x - 5 > 3(2-x)' 3) Find the ranges of values of k for which the equation x2 + (k - 3h + k : o has (a) real distinct roots, (b) roots of the same sigl' 4) If x is real and x2 + (2 - k>x + l - 2k : a show that k between certain limits, and find these limits. 5) Find the limitations required on the values of the real number that the equation x2 * Zcx - c + 2- 0 shall have real roots. (JMB)p cannot lie (JMB)P c in order (JMB)P 6) Provethat,if x2)k(x+ 1) forallreal x, then -4<k<0' (c)p 7) Find the condition that must be satisfied by k in order that the expression Zxz *6x* 1+ k(x'+2) may be positive for all real values of x ' 8) Find the range of values of x for which (JMB)p (x-4)<x(x -4)<s (Uof L)P 9) (a) If x : 2 is a root of the equation or2x2 +2(2a-5)x*8 : o find the possible value (or values) of o and the corresponding value (or values) of the other root' (b) Find the range (or ranges) of possible values of the real number a if a2x2 +2(2a-5)x+8>o for all real values of x. (c) 10) Determine for each of the three expressions f(x), s@) and h(x) the range i"i ,*g.rl .r values of x for which it is positive. Give your answers correct to i*o pti.tt of decimals. Explain, briefly, the reasons for your answers' (a) f(x)=x2*4x-6 (b) s(x) = -x'-8x +2 (C)P 11) (a) state the range of values of x for which 2x2 * 5x -12 is negative.
- 15. Funetions 57 (b) The value of the constant a is such that the quadratic function f(x) = x2 + 4x + a+ 3 is never negative. Determine the nature of the roots of the equation af(x) : (x2 +2)(a-l). Deduce the value of a for which this equation has equal roots. (AEB) "3p L2 By eliminating x and, y from the equations !+l=t, x+y:o,!-:m xyx where a*O, obtain a relation between m and a. Given that a is real, determine the ranges of values of a for which m is real. (JMB)p 13) If a) 0, prove that the quadratic expression axz + bx + c is positive for all real values of x when b2 I 4ac. Hence find the range of values of p for which the quadratic function of x I@) = 4x2 +4Px-(3P'+aP-3) is positive for all real values of x. Illustrate your result by making sketch graphs of /(x) for each of the cases p=0 and p:1. (UofL) la) (a) If a is a positive constant, find the set of values of x for which a(xz * 2x - 8) is negative. Find the value of a if this function has a minimum value of - 27. (b) Find two quadratic functions of x which arc zeto at .x = l, which take the value 10 when x : 0 and which have a maximum value of 18. Sketchthegraphsofthesetwofunctions. (Uofl-) 1 5) Find the set of values of k for which f(x) = 3x2 - 5x - k is greater than unity for all real values of x. Show that, for all k, the minimum value of /(x) occurs when ., = *. Find k if this minimum value is zero. (U of L)p 16) Provethat 3x2 -4x +2>0 forallrealvaluesof x. (U of L)p 17) Find the value of k(* l) such that the quadratic function of x k(x + 2)2 - (x - r)(x -2) is equal to zero for only one value of x. Find also (a) the range of values of k for which the function possesses a mini- mum value, (b) the range of values of k .for which the value of the function never exceeds 12.5. Sketch the graph of the function for k : * and for k :2L. (U of L) 18) Find the set of values of x for which ?x2+3x+ Z> 4 @ of L)
- 16. 58 Mathematics - The Core Course for A-level 19) The roots of the equation 9x2*6x*l:4kx where k is a real constant, are denoted by a and P. (a) Show that the equation whose roots are I la and 1/0 is x2+ 6x*g - 4kx (b) Find the set of values of k for which a and P are real. (c) Find also the set of values of k for which a and P are real 20) The function f is given by f : x * x2-3x-4, where Find the range of f and the values of x for which f{x): Q. and positive. (U of L) x€IR. (c)p SOME OTHER SIMPLE FUNCTIONS We will now consider the graphical representation of some other simple functions. A full analysis of the general form of these functions is not possible at this stage, but enough information can be deduced to give a rough idea ofthe shapes of the curves that represent them. Exponential Functions fui exponential function is one where the variable appears as an exponent, (i.e. an index) e.g. 2* , 3-x, lo'*" 5-3* + 2 are all exponential functions of x. Consider the function f(x) = 2* for which the following table shows cor- responding values of x and f(x). From this table we see that: 1. f(x) > 0 for all real values of x, 2. as x increases f(x) increases at a rapidly accelerating rate, 3.f(x)- I when x:0 4. as x decreases (i.e. x 10, - 100, . . .) f(x) quickly becomes numerically smaller and we say that, as x approaches minus infinity , f(x) approaches the value zero. This is written, x +- oo , f{x)-+0 From these observations the sketch of f(x) : 2* is drawn:
- 17. Functions 59 IW) 4 3 2 I The curve approaches the negative x-axis but never actually touches it or crosses it and we say that the x-axis is an asymptote to the curve. Another way of expressing the behaviour of /(x) for negative values of x is based on the fact that f(x) approaches a limiting value (or limit) of zero as x approaches minus infinity. i.e. the limit of /(x) as x -+- oo is zero. This property is written ,iI -t/(x)l -o fury function of the form o* , where a ) I , is represented by a curve similar to that deduced for 2* . Rational Functions A function where both numerator and denominator are polynomials is called a rational function. e.g. are rational functions of x. Consider the function f{x) values for x and f(x) f(x)- I x x 2x2 -7x x2 -l' x* I and the following table of corresponding I ) x I x -2 -l _1 2 -l r0 1 4 I 3 1 2 I 10 From this table we see that: 1. For x ) 0, ftx) > 0 and as .x -*oo, f(x) + Q. 2. For x(0, f(x)<0 andas x+ , f(x)*0 i.e. x and f(x) have the same sign.
- 18. -l J. 60 we see that as x decreases to zero, and as x increases to zeto, From these observations we can 1 f(x): - . x This curve has two asymptotes, the All the curves looked at so far have has a 'break' or discontinuity at the undefined. Mathematics - The Core Course for A'level For r : 0, /(0) :; which has no finite value and is said undefined.In such circumstances we investigate the behaviour approaches zeto. Now x can approach zerc in two ways i.e. decrease from positive values towards zero (approach zero or increase from negative values towards zero (approach zero to be of /(x) as x from above) from below) , From (1) and (2) above we see that f(x) and x have the same sign. As x decreases, f(x) increases. From the following table f(x) -) oo f(x) -.> - oo. draw the following sketch representing horizontal and vertical axes. been unbroken or continuous, but this curve point where x : 0 and /(0) is f(x
- 19. .ryLL:] :o and ,ir".[f lr I 1 but lim I t I does not have a unique value as approa x+olrj ' x depending on whether x approaches zero from above or bel say that , rgxL:] does not exist. In general, for ,rT, t/(x)l to exist with a value k, say, thenas x+a fropabove) f@+ft and as x + a from below J Functions 61 In general, if f(x) is any function of x and if /(x) is undefined for a finite value of x , x - o soy, then the curve representing f{x) has a discontinuity where x : o. Also, w€ see that _0 oo Of In this case we LOGARITHMIC FUNCTIONS For values of a)0, any function of the form logox, lo&(bc+ 1),... is a logarithmic function. Consider the function f(x) = logzx and the following table of.corresponding values for x and f{x). From this table we see that: 1. when x - -_ 1, f(- 1) : logz(- 7) : b say.But there is no real value of b for which 2b : - 1. So /(- 1) does not exist, ffid all negative values of x lead to the same conclusion, i;e,,; .ft*1,r:*:,.....16g.;* does,not exist lfOil n0gati+e,,.+*ueS..:.of1,.,:,i, 2. for x)1, f(x)>0 andas x-+oo, f(x)-+oo, 3. for x : 0, /(0) is undefined so we investigate the behaviour of f(x) as x -+ 0 from above, From the table we see that as x decreases to zero, 0<x( L, f(x) <0. ches ow. f(x) -+ - "o and that for
- 20. logzx 62 Mathematics - The Core Course for A{evel From these observations we can now sketch the graphical representation of f(x) = logzx. Any function of the form logox will have a graph of similar shape. Note that, while logx does not exist for negative values of x, the value of logox con be negative. SIMPLE VARIATIONS OF FUNCTIONS The curve on the right is one with which we are now familiar and represents the function f(x) = 2x. We will now consider how to obtain the graphs of some simple variations of this function. 1) g(x) = 2x + 1. Comparing f(x) = 2x and g(x) = 2'* + I we see that, for any one value of x , S(x) is one unit greater than f{x). So we deduce that the curve representing S(x) is the same shape as that representing f(x) but is raised vertically by one unit: f(x)
- 21. s(x) 1.e. Functions 63 / - -AsymptoteJ-'-'---''a'J In general, if the curve representing any function f(x) is known, the curve representing f(x) + c is the sirme shape but is raised vertically by c units. 2) g(x) = 2-x. Comparing S(x) = 2-x with f(x) = 2x we see that for S(x), a, input x : o gives an output value of 2-a while for f{x)) aninput x: -a gives the same output value of 2-a s@) = f{-a) or g(x) = f(-x) for x e IRi.e. So the curye representing S(x) is the same as that representing f(x) except that the negative and positive values of x are transposed: In general, for any function /(x), the curve representtng fex) is the reflection in the vertical axis of the gaph of /(x). 3) g(x) = -2x. Comparing S(x) = -2x with f(x) = 2x we see that S(x) = -f(x) so the curve representing g(x) is the same shape as that for f{x) with the positive and negative values of f(x) transposed. The vertical axis is returned to its conventional position by a reflection in the horizontal axis giving f(x) g(x) s@)
- 22. f(x) 64 Mathematics - The Core Course for A-level In general the curve representing -f(x) is the reflection in the horizontal axis of the curve representing f(x). EXERCISE 3e 1) Write down the values of f(x) = (+)* corresponding to x : 2,4,6 and to x - -2,-4,-6. From these values deduce the behaviour of f(x) as x -+ oo and as 7s + - oo. sketch the graph of f(x) = (i)* , marking any asymptotes. 2) Draw sketch graphs of the following functions: (u) 3* (b) 2x - t (r) 3-x (d) 22* (r) t + 2* (f) - (2*) showing clearly, in each case , afiy asymptotes. 3) For what value of x is f(x)= --l- undefined? Describe the behaviour I -x of /(x) as x approaches this value from above and from below. write down also, *sa t/(x)] and , jI _ [/(x)l . Use this information to sketch the graph of f(x)= --l- marking the I -X asymptotes clearly. a) By following a procedure similar to that indicated in (3), or otherwise, draw sketch graphs of the following functions: I I 2 I x-l x x-/, xf I x x (f)r+ 5) Sketch the curve representing f(x) = x2 -4. the graphs of the following functions (a) x2 -z (b) x2 + z (c) 4 -x2 On the same axes sketch g(x) g(x) = -2x
- 23. Functions 65 6) Sketch the graph of the function f(x) = le x . Also make sketch graphs of the following functions (a) ls(-x) (b) -lgx (c) l+lgx (d) l-lgx 7) Sketch the graph of the function f(x) = 2-3x and on the same set of axes sketch the graph of 3x-2. 8) On the same set of axes draw sketch graphs of the functions /(x) = 1g1 and g(x) = t0". Describe how the second graph can be obtained from the first graph. 9) Repeat Question 8 for the functions f(*) = X -t and g(x) = i@+ t). INVERSE FUNCTIONS Consider the mapping f :x + 2x, x e {2,3,4lt. Under this function the domain {23,41 maps to the image-set {4,6,8}. It is possible to reverse this mapping, i.e. we can map each member of the image-set {4,6,8} back to the corresponding member of the domain by halving each member of the image-set , i.e. {+2, 6+3, 8+4 We can also express this procedure as an algebraic relationship, i.e. if xe {4,6,8} then x-ix maps 4-+2, 6'+3, 8+4 This reverse mapping is a one-one mapping so it is a function in its own right and it is called the inverse function of / where f: x '+ 2x. Denoting this inverse function by f-' we see that f-r; x * x, x € {4,6,8} reverses the mapping f:x-+2x, xe (2,3,41. ln fact for all real values of x, f: x + 2x can be reversed. So we can say that if /isthefunctiondefinedby f:x+2sc, x€IR then f-r is the function which reverses this mapping and which is defined by f-r x-lx, x€lR.
- 24. f(x) 66 Mathematics - The Core Course for A-leyel The graphs of f and f-r are shown below. Now consider the function f: x + x2, x e This is a many-one mapping, i.e. there are two values of x which map to one value of /(x). For example both 2 and -2 map to 4. IR. f(x) We can reyerse this mapping by taking the positive and negative square root of each member of the image-set. Expressing this in algebraic form gives the mapping y + *t/x. However this is a one-m arry mapping so it is not a function and we say that f : x * x2 for x e IR does not have an inverse function. If, however, we restrict the domain of f to x e R*, i.e. we redefine our function as f : x + x2, x e IR+ then this becomes a one-one mapping. f -'(x) f (x)
- 25. Functions 67 We can now express the reverse mapping as 1s + Jx which is a one-one mapping. Sothefunction f:x+x2, xe IR* does have an inverse, viz. f-r: .x + t/x, x € IR* To summarise, a given function f maps the domain of f to the image-set of f . If the reverse mapping of the image-set of f to the dornain of f is a one-one mapping it is called the inverse function and is denoted by f-'. Note that if f defines a one-one mapping then f-r usually exists, but if f defines a many-one mapping then f-r does not exist. THE GRAPHS OF FUNCTIONS AND THEIR INVERSES The graphs of both f(x) - bc and f -'(x): *x for x e IR, and both f{x) = x2 and f -'(x}: x for x e IR* have been sketched above. Observing these we see that in each case the graph of .ft(x) is the reflection in the line AB of the graph of /(x). This is true for the graph of any function / and its inverse, /-1. (The line AB is the graph of f :x +x and this function is its own inverse, i.e. f(x) = f-r(x) in this case.) So if the graph representing a function is known, the graph representing its inverse can be sketched. It must be appreciated, however, that even when a function / does not possess an inverSe, the curve representing / can be reflected in the line AB but in this case the reflected graph does nof correspond to f-r. EXAMPLE 3f Given the function f(x) : 2*, sketch the graph of f-t " x e IR, find f-t as a function of x and * f(x) = 2x maPS x to 2*. To find f-r we have to reverse this process, i.e. map values of 2* back to values of x. If 2* = w soy, then taking logarithms to base 2 on each side gives f-'(x) a X _ LO$1W
- 26. 68 Matfiematics - The Core Course for A-level Hence the relationship can be expressed as 2x+x w + log2w The relationship w + log2w is a one-one mapping for rar e IR so it is a function and it reverses the mapping x + 2x. Replacing the variable w by the variable x we have f(x)=2x + f"(x)=lagzx EXE RCTSE 3f , l) Determine which of the following functions have an inverse for x € IR. If f-r exists, express it as a fun-ction of x. (a) 3x (b) ?x + t (c) x2 4 (d) 1og5x 2) On the same set of ixes draw a sketch graph of each function given in Question I with, where it exists, the inverse function. 3) State the inverse function, with its domaitr, of each of the functions given below. (a) f:x**x-3, x€lR (b) f:x+5x, xe{-t,0,1} (c) f:x-+5x, xeN (d) f:x*(x-1)(x-3), xcIR, x22 (e) f:x*(x-2)(x*2), xe IR* (0 f:x+ 10*, x€lR* f(x) : 2x f-'(x) : logr,

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