Math

1. RATIONAL EQUATION,
RATIONAL INEQUALITY
AND
RATIONAL FUNCTION

2. RATIONAL EQUATION
- An equation involving rational expression.
Example:
𝑥 − 2
𝑥 + 3
=
5
2
𝑥3
+ 1
5
=
𝑥
2
6
𝑥 + 3
=
8
3

3. RATIONAL INEQUALITY
- An equality involving rational expression.
( < , >, ≥, ≤ )
Example:
𝑥
𝑥 − 1
<
5
𝑥
4
𝑥 + 9
>
5 − 𝑥
𝑥
𝑥2
− 3𝑥
3
≥
𝑥
2

4. RATIONAL FUNCTION
- A function of the form f(x) =
𝑝(𝑥)
𝑞(𝑥)
, where p(x) and q(x)
are polynomial functions and q(x) is not the zero of function.
Example:
𝑓 𝑥 =
5 + 𝑥
𝑥 − 2
𝑓 𝑥 =
𝑥3
+ 𝑥
𝑥 − 2
𝑥2
+ 1
5
= 𝑦

5. Let’s try this!
1.
4
3𝑥 + 1
<
5
𝑥
2.
𝑥
𝑥 − 1
= y
5.
4𝑥 − 2
3𝑥
=
5
𝑥
3.
5𝑥
6
=
1
𝑥
4.
7
8𝑥 + 2
>
5
𝑥 Rational Inequality
Rational Inequality
Rational Function
Rational Equation
Not of these

6. INTERCEPTS, ZEROES
AND ASYMPTOTES
OF
RATIONAL FUNCTION

7. INTERCEPTS
- Are points of intersection of its graph and an axis .
X-INTERCEPTS
- Occurs at the zeros of the numerator
that are not zeros of denominator.
Y-INTERCEPTS
- Occurs at f(0), provided that f(x) is
defined at x = 0.
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8. DETERMINING X-INTERCEPTS AND Y-INTERCEPTS.
X-INTERCEPTS. Equate the numerator to 0.
𝑓 𝑥 =
5 + 𝑥
𝑥 − 2
Example:

9. DETERMINING X-INTERCEPTS AND Y-INTERCEPTS.
X-INTERCEPTS. Equate the numerator to 0.
𝑓 𝑥 =
3𝑥
𝑥 + 3
Example:

10. DETERMINING X-INTERCEPTS AND Y-INTERCEPTS.
X-INTERCEPTS. Equate the numerator to 0.
𝑓 𝑥 =
𝑥 − 2
𝑥 + 6
Example:

11. DETERMINING X-INTERCEPTS AND Y-INTERCEPTS.
Y-INTERCEPTS. Evaluate the function at x = 0.
𝑓 𝑥 =
5 + 𝑥
𝑥 − 2
Example:

12. DETERMINING X-INTERCEPTS AND Y-INTERCEPTS.
𝑓 𝑥 =
3𝑥
𝑥 + 3
Example:
Y-INTERCEPTS. Evaluate the function at x = 0.

13. DETERMINING X-INTERCEPTS AND Y-INTERCEPTS.
𝑓 𝑥 =
𝑥 − 2
𝑥 + 6
Example:
Y-INTERCEPTS. Evaluate the function at x = 0.

14. Let’s try this!
Determine the intercepts of the following.
𝑓 𝑥 =
3 − 𝑥
𝑥 − 9
𝑓 𝑥 =
7𝑥 − 3
𝑥 − 2

15. ZEROES OF RATIONAL FUNCTION
- Values of x which make the function zero..
𝑓 𝑥 =
2𝑥 − 1
𝑥 + 9
Example:

16. ZEROES OF RATIONAL FUNCTION
- Values of x which make the function zero..
𝑓 𝑥 =
5
𝑥 + 9
Example:

17. Let’s try this!
Determine the intercepts of the following.
𝑓 𝑥 =
𝑥 − 6
𝑥 − 1
𝑓 𝑥 =
𝑥 − 12
4 + 3

18. ASYMPTOTES OF RATIONAL FUNCTION
- An imaginary line to which a graph gets and closer as
the x or y increases or decreases its value without limit.
https://courses.lumenlearning.com/

19. ASYMPTOTES OF RATIONAL FUNCTION
- An imaginary line to which a graph gets and closer as
the x or y increases or decreases its value without limit.
https://sciencetrends.com/

20. ASYMPTOTES OF RATIONAL FUNCTION
- An imaginary line to which a graph gets and closer as
the x or y increases or decreases its value without limit.
https://courses.lumenlearning.com/