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

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

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

6. Example: 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 = 𝑥−1

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

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

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

16. Evaluating Piecewise Functions
Evaluate 𝑓 𝑥 when (a) 𝑥=0 (b) 𝑥=2 and (c) 𝑥=4 𝑓 𝑥 = 𝑥+2, 𝑖𝑓 𝑥<2 2𝑥+1, 𝑖𝑓 𝑥≥2

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 = 4, 𝑥≤−2 𝑥 2 , −2<𝑥<2 4, 𝑥≥2

20. Graphing Piecewise Functions
𝑓 𝑥 = 3𝑥+12, 𝑖𝑓 𝑥<−3 |𝑥|, 𝑖𝑓 −3≤𝑥<5 −3𝑥+12, 𝑖𝑓 𝑥≥5

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

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

24. Rules for Transformation of Functions

25. Example: Describe the Transformation and Sketch the Graph
𝑔 𝑥 = 𝑥 2 −1

26. Example: Describe the Transformation and Sketch the Graph
𝑓 𝑥 =2|𝑥−1|

27. Example: Describe the Transformation and Sketch the Graph
𝑔 𝑥 =−2 𝑥

28. Example: Describe the Transformation and Sketch the Graph
𝑔 𝑥 =−3𝑥−2

29. Example: Describe the Transformation and Sketch the Graph
𝑔 𝑥 =− 𝑥 2 +1

30. Example: Describe the Transformation and Sketch the Graph
𝑔 𝑥 =−|𝑥−2|

31. Example: write the Equation Described
Absolute Value – vertical shift up 5, horizontal shift right 3

32. Example: write the Equation Described
Linear – vertical stretch/compression by 2 5

33. Example: write the Equation Described
Quadratic – vertical stretch by 5, horizontal shift left 8, reflected over the x- axis.

34. Combination of Functions

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

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

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

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

39. Composition of Functions

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

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

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