FiveMinute Check over Chapter 5 CCSS ThenNow New

Over Chapter 5 Estimate the x-coordinates at which the relative maxima and relative minima

Over Chapter 5 Solve p 3 – 2 p 2 – 8 p =

Over Chapter 5 Solve x 4 – 7 x 2 + 12 = 0.

Over Chapter 5 Which is not a possible rational zero for the function f(x)

Over Chapter 5 Which of the following could not be a zero of the

Content Standards F. IF. 9 Compare properties of two functions each represented in a

You performed operations on polynomials. • Find the sum, difference, product, and quotient of

Add and Subtract Functions A. Given f(x) = 3 x 2 + 7 x

Add and Subtract Functions B. Given f(x) = 3 x 2 + 7 x

A. Given f(x) = 2 x 2 + 5 x + 2 and g(x)

B. Given f(x) = 2 x 2 + 5 x + 2 and g(x)

Multiply and Divide Functions A. Given f(x) = 3 x 2 – 2 x

Multiply and Divide Functions B. Given f(x) = 3 x 2 – 2 x

Multiply and Divide Functions Since 4 makes the denominator 0, it is excluded from

A. Given f(x) = 2 x 2 + 3 x – 1 and g(x)

B. Given f(x) = 2 x 2 + 3 x – 1 and g(x)

Compose Functions A. If f(x) = (2, 6), (9, 4), (7, 7), (0, –

Compose Functions To find g ○ f, evaluate f(x) first. Then use the range

Compose Functions B. Find [f ○ g](x) and [g ○ f](x) for f(x) =

Compose Functions = 3(4 x 2 – 4 x + 1) – 2 x

Compose Functions Answer: = 2(3 x 2 – x + 4) – 1 Substitute

Compose Functions = 2(3 x 2 – x + 4) – 1 Substitute 3

B. Find [f ○ g](x) and [g ○ f](x) for f(x) = x 2

Use Composition of Functions TAXES Hector has $100 deducted from every paycheck for retirement.

Use Composition of Functions Plan Write equations for r(x) and t(x). $100 is deducted

Use Composition of Functions = 1400 – 0. 04(1400) Replace x with 1400 in

Use Composition of Functions = 1440 – 100 = 1340 Answer: Replace x with

Use Composition of Functions = 1440 – 100 Replace x with 1440 in r(x)

TAXES Brandi Smith has $200 deducted from every paycheck for retirement. She can have

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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

Over Chapter 5 Estimate the x-coordinates at which the relative maxima and relative minima occur for the graph of f(x) = x 4 + 16 x 2 – 25. A. 0 B. 1 C. 4 D. 5

Over Chapter 5 Solve p 3 – 2 p 2 – 8 p = 0. A. – 2, 2 B. – 2, 0, 4 C. 0, 2, 6 D. 2, 4, 6

Over Chapter 5 Solve x 4 – 7 x 2 + 12 = 0. A. B. C. D.

Over Chapter 5 Which is not a possible rational zero for the function f(x) = 3 x 3 – 12 x 2 – 5 x + 15? A. ± 15 B. C. D.

Over Chapter 5 Which of the following could not be a zero of the function g(x) = 4 x 3 – 5 x + 26? A. B. 1 C. 4 D. 13

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.

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

• composition of functions

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

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

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

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

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

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

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

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

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

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

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

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

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:

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) 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:

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

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

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

Compose Functions = 2(3 x 2 – x + 4) – 1 Substitute 3 x 2 – x + 4 for x in g(x). = 6 x 2 – 2 x + 7 Simplify. Answer: So, [f ○ g](x) = 12 x 2 – 14 x + 8; D = {all real numbers}, R = {y│y > 3. 91}; and [g ○ f](x) = 6 x 2 – 2 x + 7; D = {all real numbers}, R = {y│y > 6. 33}.

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

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.

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%. Solve t(x) = x – 0. 04 x If Hector has his retirement deducted before taxes, then his net income is represented by [t ○ r](1500)= t(1500 – 100) = t(1400) Replace x with 1500 in r(x) = x – 100.

Use Composition of Functions = 1400 – 0. 04(1400) Replace x with 1400 in t(x) = x – 0. 04 x. = 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. 04 x. [r ○ t](1500) = r[1500 – 0. 04(1500)] = r(1500 – 60) = r(1440)

Use Composition of Functions = 1440 – 100 = 1340 Answer: 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.

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.