1. Splash Screen

2. Five-Minute Check (over Chapter 5) CCSS Then/Now New Vocabulary
Key Concept: Operations on Functions
Example 1: Add and Subtract Functions
Example 2: Multiply and Divide Functions
Key Concept: Composition of Functions
Example 3: Compose Functions
Example 4: Real-World Example: Use Composition of Functions
Lesson Menu

3. Estimate the x-coordinates at which the relative maxima and relative minima occur for the graph of f(x) = x4 + 16x2 – 25.
A. 0
B. 1
C. 4
D. 5
5-Minute Check 1

4. Solve p3 – 2p2 – 8p = 0.
A. –2, 2
B. –2, 0, 4
C. 0, 2, 6
D. 2, 4, 6
5-Minute Check 2

5. Solve x4 – 7x = 0. A. B. C. D.
5-Minute Check 3

6. Which is not a possible rational zero for the function f(x) = 3x3 – 12x2 – 5x + 15?D.
5-Minute Check 4

7. Which of the following could not be a zero of the function g(x) = 4x3 – 5x + 26?
5-Minute Check 5

8. F.BF.1.b Combine standard function types using arithmetic operations.
Content Standards
F.IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
F.BF.1.b Combine standard function types using arithmetic operations.
Mathematical Practices
2 Reason abstractly and quantitatively.
CCSS

9. You performed operations on polynomials.
Find the sum, difference, product, and quotient of functions.
Find the composition of functions.
Then/Now

10. composition of functions
Vocabulary

11. Concept

12. A. Given f(x) = 3x2 + 7x and g(x) = 2x2 – x – 1, find (f + g)(x).
Add and Subtract Functions
A. Given f(x) = 3x2 + 7x and g(x) = 2x2 – x – 1, find (f + g)(x).
(f + g)(x) = f(x) + g(x) Addition of functions
= (3x2 + 7x) + (2x2 – x – 1) f(x) = 3x2 + 7x and g(x) = 2x2 – x – 1
= 5x2 + 6x – 1 Simplify.
Answer: 5x2 + 6x – 1
Example 1

13. B. Given f(x) = 3x2 + 7x and g(x) = 2x2 – x – 1, find (f – g)(x).
Add and Subtract Functions
B. Given f(x) = 3x2 + 7x and g(x) = 2x2 – x – 1, find (f – g)(x).
(f – g)(x) = f(x) – g(x) Subtraction of functions
= (3x2 + 7x) – (2x2 – x – 1) f(x) = 3x2 + 7x and g(x) = 2x2 – x – 1
= x2 + 8x + 1 Simplify.
Answer: x2 + 8x + 1
Example 1

14. A. Given f(x) = 2x2 + 5x + 2 and g(x) = 3x2 + 3x – 4, find (f + g)(x).
A. 5x2 + 8x – 2
B. 5x2 + 8x + 6
C. x2 – 2x – 6
D. 5x4 + 8x2 – 2
Example 1

15. B. Given f(x) = 2x2 + 5x + 2 and g(x) = 3x2 + 3x – 4, find (f – g)(x).
A. –x2 + 2x + 5
B. x2 – 2x – 6
C. –x2 + 2x – 2
D. –x2 + 2x + 6
Example 1

16. A. Given f(x) = 3x2 – 2x + 1 and g(x) = x – 4, find (f ● g)(x).
Multiply and Divide Functions
A. Given f(x) = 3x2 – 2x + 1 and g(x) = x – 4, find (f ● g)(x).
(f ● g)(x) = f(x) ● g(x) Product of functions
= (3x2 – 2x + 1)(x – 4) Substitute.
= 3x2(x – 4) – 2x(x – 4) + 1(x – 4) Distributive Property
= 3x3 – 12x2 – 2x2 + 8x + x – 4 Distributive Property
= 3x3 – 14x2 + 9x – 4 Simplify.
Answer: 3x3 – 14x2 + 9x – 4
Example 2

17. B. Given f(x) = 3x2 – 2x + 1 and g(x) = x – 4, find
Multiply and Divide Functions
B. Given f(x) = 3x2 – 2x + 1 and g(x) = x – 4, find
Division of functions
f(x) = 3x2 – 2x + 1 and g(x) = x – 4
Answer:
Example 2

18. Since 4 makes the denominator 0, it is excluded from the domain of
Multiply and Divide Functions
Since 4 makes the denominator 0, it is excluded from the domain of
Example 2

19. A. Given f(x) = 2x2 + 3x – 1 and g(x) = x + 2, find (f ● g)(x).
A. 2x3 + 3x2 – x + 2
B. 2x3 + 3x – 2
C. 2x3 + 7x2 + 5x – 2
D. 2x3 + 7x2 + 7x + 2
Example 2

20. B. Given f(x) = 2x2 + 3x – 1 and g(x) = x + 2, find .C. D.
Example 2

21. Concept

22. Answer: f ○ g = {(7, –1), (–1, 7), (4, 4), (8, 6)}
Compose Functions
A. If f(x) = (2, 6), (9, 4), (7, 7), (0, –1) and g(x) = (7, 0), (–1, 7), (4, 9), (8, 2), find [f ○ g](x) and [g ○ f](x).
To find f ○ g, evaluate g(x) first. Then use the range of g as the domain of f and evaluate f(x).
f[g(7)] = f(0) or –1 g(7) = 0
f[g(–1)] = f(7) or 7 g(–1) = 7
f[g(4)] = f(9) or 4 g(4) = 9
f[g(8)] = f(2) or 6 g(8) = 2
Answer: f ○ g = {(7, –1), (–1, 7), (4, 4), (8, 6)}
Example 3

23. g[f(2)] = g(6) g(6) is undefined. g[f(9)] = g(4) or 9 f(9) = 4
Compose Functions
To find g ○ f, evaluate f(x) first. Then use the range of f as the domain of g and evaluate g(x).
g[f(2)] = g(6) g(6) is undefined.
g[f(9)] = g(4) or 9 f(9) = 4
g[f(7)] = g(7) or 0 f(7) = 7
g[f(0)] = g(–1) or 7 f(0) = –1
Answer: Since 6 is not in the domain of g, g ○ f is undefined for x = 2. g ○ f = {(9, 9), (7, 0), (0, 7)}
Example 3

24. [f ○ g](x) = f[g(x)] Composition of functions
Compose Functions
B. Find [f ○ g](x) and [g ○ f](x) for f(x) = 3x2 – x + 4 and g(x) = 2x – 1. State the domain and range for each combined function.
[f ○ g](x) = f[g(x)] Composition of functions
= f(2x – 1) Replace g(x) with 2x – 1.
= 3(2x – 1)2 – (2x – 1) + 4 Substitute 2x – for x in f(x).
Example 3

25. = 3(4x2 – 4x + 1) – 2x + 1 + 4 Evaluate (2x – 1)2.
Compose Functions
= 3(4x2 – 4x + 1) – 2x Evaluate (2x – 1)2.
= 12x2 – 14x + 8 Simplify.
[g ○ f](x) = g[f(x)] Composition of functions
= g(3x2 – x + 4) Replace f(x) with x2 – x + 4.
Example 3

26. = 2(3x2 – x + 4) – 1 Substitute 3x2 – x + 4 for x in g(x).
Compose Functions
= 2(3x2 – x + 4) – 1 Substitute 3x2 – x for x in g(x).
= 6x2 – 2x + 7 Simplify.
Answer: So, [f ○ g](x) = 12x2 – 14x + 8; D = {all real numbers}, R = {y│y > 3.91}; and [g ○ f](x) = 6x2 – 2x + 7; D = {all real numbers}, R = {y│y > 6.33}.
Example 3

27. A. If f(x) = {(1, 2), (0, –3), (6, 5), (2, 1)} and g(x) = {(2, 0), (–3, 6), (1, 0), (6, 7)}, find f ○ g and g ○ f.
A. f ○ g = {(2, –3), (–3, 5), (1, –3)}; g ○ f = {(1, 0), (0, 6), (2, 0)}
B. f ○ g = {(1, 0), (0, 6), (2, 0)}; g ○ f = {(2, –3), (–3, 5), (1, –3)}
C. f ○ g = {(–3, 2), (5, –3), (–3, 1)}; g ○ f = {(0, 1), (6, 0), (0, 2)}
D. f ○ g = {(0, 1), (6, 0), (0, 2)}; g ○ f = {(–3, 2), (5, –3), (–3, 1)}
Example 3

28. B. Find [f ○ g](x) and [g ○ f](x) for f(x) = x2 + 2x + 3 and g(x) = x + 5. State the domain and range for each combined function.
A. [f ○ g](x) = x2 + 2x + 8; D = {all real numbers}, R = {y│y ≥ 2}; and [g ○ f](x) = x2 + 12x + 38; D = {all real numbers}, R = {y│y ≥ 7}
B. [f ○ g](x) = x2 + 12x + 38; D = {all real numbers}, R = {y│y ≥ 7}; and [g ○ f](x) = x2 + 12x + 38; D = {all real numbers}, R = {y│y ≥ 2}
C. [f ○ g](x) = x2 + 12x + 38; D = {all real numbers}, R = {y│y ≥ 2}; and [g ○ f](x) = x2 + 2x + 8; D = {all real numbers}, R = {y│y ≥ 7}
Example 3

29. Use Composition of Functions
TAXES Hector has $100 deducted from every paycheck for retirement. He can have this deduction taken before state taxes are applied, which reduces his taxable income. His state income tax is 4%. If Hector earns $1500 every pay period, find the difference in his net income if he has the retirement deduction taken before or after state taxes.
Understand Let x = his income per paycheck, r(x) = his income after the deduction for retirement, and t(x) = his income after the deduction for state income tax.
Example 4

30. Plan Write equations for r(x) and t(x).
Use Composition of Functions
Plan Write equations for r(x) and t(x).
$100 is deducted for retirement. r(x) = x – 100
The tax rate is 4%. t(x) = x – 0.04x
Solve If Hector has his retirement deducted before taxes, then his net income is represented by [t ○ r](1500).
[t ○ r](1500) = t(1500 – 100) Replace x with in r(x) = x – 100.
= t(1400)
Example 4

31. = 1400 – 0.04(1400) Replace x with 1400 in t(x) = x – 0.04x.
Use Composition of Functions
= 1400 – 0.04(1400) Replace x with 1400 in t(x) = x – 0.04x.
= 1344
If Hector has his retirement deducted after taxes, then his net income is represented by [r ○ t](1500). Replace x with 1500 in t(x) = x – 0.04x.
[r ○ t](1500) = r[1500 – 0.04(1500)]
= r(1500 – 60)
= r(1440)
Example 4

32. = 1440 – 100 Replace x with 1440 in r(x) = x – 100.
Use Composition of Functions
= 1440 – 100 Replace x with 1440 in r(x) = x – 100.
= 1340
Answer: [t ○ r](1500) = 1344 and [r ○ t](1500) = The difference is 1344 – 1340 or 4. So, his net income is $4 more if the retirement deduction is taken before taxes.
Example 4

33. TAXES Brandi Smith has $200 deducted from every paycheck for retirement. She can have this deduction taken before state taxes are applied, which reduces her taxable income. Her state income tax is 10%. If Brandi earns $2200 every pay period, find the difference in her net income if she has the retirement deduction taken before state taxes.
A. Her net income is $20 less if she has the retirement deduction taken before her state taxes.
B. Her net income is $20 more if she has the retirement deduction taken before her state taxes.
C. Her net income is $10 less if she has the retirement deduction taken before her state taxes.
D. Her net income is $10 more if she has the retirement deduction taken before her state taxes.
Example 4

34. End of the Lesson