1. Functions and Their Graphs
Honors Calculus
Keeper 4

2. Evaluating Functions
To evaluate a function, substitute the input (the given number or expression) for the function's variable (place holder, x).  Replace the x with the number or expression

3. Example 1: Evaluating Functions
Given: 𝑓 𝑥 =3𝑥−5 Find: 𝑓(4)

4. Example 2: Evaluating Functions
Given: 𝑓 𝑥 = 𝑥 2 +7 Find: 𝑓(3𝑎)

5. Example 3: Evaluating Functions
Given: 𝑓 𝑥 = 𝑥 2 +7 Find: 𝑓(𝑏−1)

6. Example 4: Evaluating Functions
Given: 𝑓 𝑥 = 𝑥 2 +7 Find: 𝑓 𝑥+ℎ −𝑓 𝑥 ℎ

7. Domain

8. Range

9. Find the Domain and Range of the Function
𝑓 𝑥 = 𝑥−1

10. Find the Domain and Range of the Function
𝑓 𝑥 =4−𝑥

11. Find the Domain and Range of the Function
𝑓 𝑥 = 4 𝑥

12. Find the Domain and Range of the Function
𝑓 𝑥 = 1 2 𝑥 3 +2

13. Find the Domain and Range of the Function
𝑓 𝑥 = 9− 𝑥 2

15. Rules for Transformation of Functions

16. Piecewise Functions
A function that is defined using two or more equations for different intervals of the domain is called a piecewise function.

17. Evaluating Piecewise Functions
Evaluate 𝑓 𝑥 when (a) 𝑥=67 (b) 𝑥=72 and (c) 𝑥=65 𝑓 𝑥 = 1.6𝑥−41.6, 𝑖𝑓 63<𝑥<66 3𝑥−132, 𝑖𝑓 66≤𝑥≤68 2𝑥−66, 𝑖𝑓 𝑥>68

18. Graphing Piecewise Functions
𝑓 𝑥 = −3𝑥+2, 𝑥≤ 𝑥−4, 𝑥>2

19. Graphing Piecewise Functions
𝑓 𝑥 = 𝑥 2 −4, 𝑖𝑓 𝑥<−2 𝑥+6 , 𝑖𝑓 −2≤𝑥≤ 𝑥−5, 𝑖𝑓 𝑥>3

20. Example. Write the equation for the graph shown.

21. Combination of Functions

22. Example
𝑓 𝑥 = 𝑥 𝑥 𝑔 𝑥 =−3𝑥+2 1. Find (𝑔⋅𝑓)(𝑥)

23. Example
𝑓 𝑥 = 𝑥 𝑥 𝑔 𝑥 =−3𝑥+2 2. Find (𝑓+𝑔)(𝑥)

24. Example
𝑓 𝑥 = 𝑥 𝑥 𝑔 𝑥 =−3𝑥+2 3. Find (𝑓−𝑔)(𝑥)

25. Example
𝑓 𝑥 = 𝑥 𝑥 𝑔 𝑥 =−3𝑥+2 4. Find 𝑔 𝑓 (𝑥)

26. Composition of Functions

27. Example
𝑓 𝑥 = 𝑥 𝑥 𝑔 𝑥 =−3𝑥+2 5. Find (𝑔∘𝑓)(𝑥)

28. Example
𝑓 𝑥 = 𝑥 𝑥 𝑔 𝑥 =−3𝑥+2 6. Find (𝑓∘𝑔)(𝑥)