A function transforms inputs (domain) to outputs (range). The inverse of a function reverses the inputs and outputs, so the domain of the inverse is the original function's range, and the range of the inverse is the original function's domain. To have an inverse, a function must be one-to-one, meaning each input maps to a single unique output. The graph of an inverse is a reflection of the original function's graph across the line y=x.

1. Functions A function is an operation performed on an input (x) to produce an output (y = f(x) ). The Domain of f is the set of all allowable inputs (x values) The Range of f is the set of all outputs (y values) Domain Range f x y =f( x )

2. To be well defined a function must  - Have a value for each x in the domain - Have only one value for each x in the domain e.g y = f( x ) = √ (x-1), x   is not well defined as if x < 1 we will be trying to square root a negative number. y = f( x ) = 1/( x-2), x   is not well defined as if x = 2 we will be trying to divide by zero. This is not a function as some x values correspond to two y values.

3. Domain y = (x-2) 2 +3 2 The Range is f( x ) ≥ 3 Finding the Range of a function Draw a graph of the function for its given Domain The Range is the set of values on the y-axis for which a horizontal line drawn through that point would cut the graph. Range Domain 3 y = (x-2) 2 +3 The Function is f( x ) = (x-2) 2 +3 , x  Link to Inverse Functions

4. The Function is f( x ) = 3 – 2 x , x  The Range is f( x ) < 3

5. f x g(f( x )) = gf(x) g f( x ) Finding gf(x) Note : gf(x) does not mean g(x) times f(x). Note : When finding f(g(x)) Replace all the x’s in the rule for the f funcion with the expression for g(x) in a bracket. e.g If f(x) = x 2 –2x then f(x-2) = (x-2) 2 – 2(x-2) Composite Functions gf(x) means “g of f of x” i.e g(f(x)) . First we apply the f function. Then the output of the f function becomes the input for the g function. Notice that gf means f first and then g. Example if f(x) = x + 3, x  and g(x) = x 2 , x  then gf(x) = g(f(x)) = g(x + 3) = (x+3) 2 , x  fg(x) = f(g(x)) = f(x 2 ) = x 2 +3, x  g 2 (x) means g(g(x)) = g(x 2 ) = (x 2 ) 2 = x 4 , x  f 2 (x) means f(f(x)) = f(x+3) = (x+3) + 3 = x + 6 , x 

6. Notice that fg and gf are not the same. The Domain of gf is the same as the Domain of f since f is the first function to be applied. The Domain of fg is the same as the Domain of g. For gf to be properly defined the Range (output set) of f must fit inside the Domain (input set) of g. For example if g(x) = √ x , x ≥ 0 and f(x) = x – 2, x  Then gf would not be well defined as the output of f could be a negative number and this is not allowed as an input for g. However fg is well defined, fg(x) = √ x – 2, x ≥ 0.

7. Domain of f Range of f = Domain of f -1 = Range of f - 1 Note: f -1 (x) does not mean 1/f(x). Inverse Functions. The inverse of a function f is denoted by f -1 . The inverse reverses the original function. So if f(a) = b then f -1 (b) = a b f a f -1

8. One to one Functions If a function is to have an inverse which is also a function then it must be one to one . This means that a horizontal line will never cut the graph more than once. i.e we cannot have f(a) = f(b) if a ≠ b, Two different inputs (x values) are not allowed to give the same output (y value). For instance f(-2) = f(2) = 4 y = f(x) = x 2 with domain x  is not one to one. So the inverse of 4 would have two possibilities : -2 or 2. This means that the inverse is not a function. We say that the inverse function of f does not exist. If the Domain is restricted to x ≥ 0 Then the function would be one to one and its inverse would be f -1 (x) = √ x , x ≥ 0

10. Example: Find the inverse of the function y = f(x) = (x-2) 2 + 3 , x ≥ 2 Sketch the graphs of y = f(x) and y = f -1 (x) on the same axes showing the relationship between them. Domain This is the function we considered earlier except that its domain has been restricted to x ≥ 2 in order to make it one-to-one. We know that the Range of f is y ≥ 3 and so the domain of f -1 will be x ≥ 3. Note: we could also have - √ (x –3) = y-2 and y = 2 - √ (x –3) But this would not fit our function as y must be greater than 2 (see graph) Rule Swap x and y to get x = (y-2) 2 + 3 Now make y the subject x – 3 = (y-2) 2 √ (x –3) = y-2 y = 2 + √ (x –3) So Final Answer is: f -1 (x) = 2 + √ (x –3) , x ≥ 3 Graphs Reflect in y = x to get the graph of the inverse function . Note: Remember with inverse functions everything swaps over. Input and output (x and y) swap over Domain and Range swap over Reflecting in y = x swaps over the coordinates of a point so (a,b) on one graph becomes (b,a) on the other.