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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

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

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

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

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

Solve x4 – 7x2 + 12 = 0. A. B. C. D. 5-Minute Check 3

Solve x4 – 7x2 + 12 = 0. A. B. C. D. 5-Minute Check 3

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

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

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

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

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

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

composition of functions Vocabulary

Concept

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: Example 1

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

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: Example 1

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

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

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

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

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

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: Example 2

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

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

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

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

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

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

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

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

Concept

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: Example 3

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

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: Example 3

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

[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 – 1 for x in f(x). Example 3

= 3(4x2 – 4x + 1) – 2x + 1 + 4 Evaluate (2x – 1)2. Compose Functions = 3(4x2 – 4x + 1) – 2x + 1 + 4 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 3x2 – x + 4. Example 3

= 2(3x2 – x + 4) – 1 Substitute 3x2 – x + 4 for x in g(x). Compose Functions = 2(3x2 – x + 4) – 1 Substitute 3x2 – x + 4 for x in g(x). = 6x2 – 2x + 7 Simplify. Answer: Example 3

= 2(3x2 – x + 4) – 1 Substitute 3x2 – x + 4 for x in g(x). Compose Functions = 2(3x2 – x + 4) – 1 Substitute 3x2 – x + 4 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

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

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

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

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

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

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 1500 in r(x) = x – 100. = t(1400) Example 4

= 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

= 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: Example 4

= 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) = 1340. The difference is 1344 – 1340 or 4. So, his net income is $4 more if the retirement deduction is taken before taxes. Example 4

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

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

End of the Lesson