1. Evaluating Functions and Difference Quotient

2. Objectives
Standard 24.0
I will evaluate the value of any functions, including piecewise functions.
Calculus Standard 4.0
I will evaluate the difference quotient of any given function.

3. When we evaluate a function we are finding the function value for
a specific input. To do this we replace the function variable in
the function’s formula with the specific input and proceed from
there.
The “specific input” can be a constant, another variable or an
algebraic expression. The important thing to remember is it replaces
the function variable everywhere in the function’s formula
For example:

4. Another example:
Because
is an imaginary number 1 is not in the domain
of f (x)

5. Another example:

6. Another example:
Because 5/0 is undefined 4 is not in the domain of h(x).

7. Problems - 1
Given
find

8. Problems - 2

9. Sometimes a function has different rules or formulas depending
on what the input value is. These functions are known as
piece-wise defined functions.

10. Problems - 3

11. The Difference Quotient
The difference quotient of a function f (x) is defined as follows:
This is used in calculus when finding derivatives so it is
worthwhile to become familiar with it in precalculus.

12. As we take Q closer to P, the accuracy with which the slope of the secant line approximates the slope of the tangent line increases.

13. Find the difference quotients for the following functions:

14. The difference quotient for

15. The difference quotient for

16. The difference quotient for

17. Problems - 4
Find the difference quotient for the following function (click on mouse to see answer).

18. Problems - 5
Find the difference quotient for the following function (click on mouse to see answer).

19. Practice
Textbook P.69 Q Answers

20. Problems - 6
Find the difference quotient for the following function (click on mouse to see answer).

21. Average Rate of Change
A special kind of difference quotient is the average rate of change. We can use the function values at two different points, a and b to find the average rate of change of a function over the interval [ a, b ]. This is given by:
Notice, this is equal to the slope of the line connecting the two points
( a, f(a) ) and ( b, f(b) ).
The average rate of change for the function f (x) = x2 over the interval [2,4] is:

22. Find the average rate of change for each of the following functions over the given intervals (click on mouse to see answer).