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 andg(x) = 2x2 – x –

1 = 5x2 + 6x – 1 Simplify.

Answer: 5x2 + 6x – 1

Add and Subtract Functions

B. Given f(x) = 3x2 + 7x and g(x) = 2x2 – x – 1, find (f – g)(x).

= x2 + 8x + 1 Simplify.

(f – g)(x) = f(x) – g(x) Subtraction of functions

= (3x2 + 7x) – (2x2 – x – 1) f(x) = 3x2 + 7x andg(x) = 2x2 – x –

1

Answer: x2 + 8x + 1

A. 5x2 + 8x – 2

B. 5x2 + 8x + 6

C. x2 – 2x – 6

D. 5x4 + 8x2 – 2

A. 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

B. Given f(x) = 2x2 + 5x + 2 and g(x) = 3x2 + 3x – 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 – 4Distributive Property

= 3x3 – 14x2 + 9x – 4Simplify.

Answer: 3x3 – 14x2 + 9x – 4

Multiply and Divide Functions

Division of functions

B. Given f(x) = 3x2 – 2x + 1 and g(x) = x – 4, find

f(x) = 3x2 – 2x + 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. 2x3 + 3x2 – x + 2

B. 2x3 + 3x – 2

C. 2x3 + 7x2 + 5x – 2

D. 2x3 + 7x2 + 7x + 2

A. Given f(x) = 2x2 + 3x – 1 and g(x) = x + 2, find (f ● g)(x).

B. Given f(x) = 2x2 + 3x – 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: f ○ g = {(7, –1), (–1, 7), (4, 4), (8, 6)}

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).

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)}

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

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).

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.

Compose Functions

= 2(3x2 – x + 4) – 1 Substitute3x2 – x + 4for 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}.

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)}

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](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}

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.

Homework

p. 389 # 1, 3, 5, 9 – 48 (x3)

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%. 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 inr(x) = x – 100.

= t(1400)

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 int(x) = x – 0.04x.

[r ○ t](1500) = r[1500 – 0.04(1500)]

= r(1500 – 60)

= r(1440)

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.

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.

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.