Precalculus

9th Edition

ISBN: 9780321716835

Author: Michael Sullivan

Publisher: Addison Wesley

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Question

Chapter 2, Problem 17RE

To determine

The value of $(f+g)(x)$ and the domain.

Expert Solution

Answer to Problem 17RE

The value of $(f+g)(x)$ is $3+2x$ .

Domain of $(f+g)(x)$ is set of all real numbers.

Explanation of Solution

Given information:

The functions are $f(x)=2−x$ and $g(x)=3x+1$ .

Formula Used:

Let f and g be two functions.

Use the sum formula, $(f+g)(x)=f(x)+g(x)$

For a composite function $f+g$ , the domain of the function is defined by

$(Domain of f)∩(Domain of g)$

If domain is not specified for a function, one should consider the set of largest real number where the function is defined all over.

Calculation:

The functions are $f(x)=2−x$ and $g(x)=3x+1$ .

Use the sum formula, $(f+g)(x)=f(x)+g(x)$ .

Substitute $2−x$ for $f(x)$ and $3x+1$ for $g(x)$ in the sum formula.

$(f+g)(x)=f(x)+g(x)=2−x+3x+1=3+2x$

Hence, the value of $(f+g)(x)$ is $3+2x$ .

As domain of $f(x)=2−x$ and $g(x)=3x+1$ is set of real numbers.

So, domain of $f+g$ is set of all real numbers.

To determine

The value of $(f−g)(x)$ and its domain.

Expert Solution

Answer to Problem 17RE

The value of $(f−g)(x)$ is $1−4x$ .

Domain of $(f−g)(x)$ is set of all real numbers.

Explanation of Solution

Given information:

The functions are $f(x)=2−x$ and $g(x)=3x+1$ .

Formula Used:

Let f and g be two functions.

Use the difference formula, $(f−g)(x)=f(x)−g(x)$ .

For a composite function $f−g$ , the domain of the function is defined by

$(Domain of f)∩(Domain of g)$

If domain is not specified for a function, one should consider the set of largest real number where the function is defined all over.

Calculation:

The functions are $f(x)=2−x$ and $g(x)=3x+1$ .

Use the difference formula, $(f−g)(x)=f(x)−g(x)$ .

Substitute $2−x$ for $f(x)$ and $3x+1$ for $g(x)$ in the difference formula.

$(f−g)(x)=f(x)−g(x)=2−x−3x−1=1−4x$

Hence, the value of $(f−g)(x)$ is $1−4x$ .

As domain of $f(x)=2−x$ and $g(x)=3x+1$ is set of real numbers.

So, domain of $f−g$ is set of all real numbers.

To determine

The value of $(fg)(x)$ and its domain.

Expert Solution

Answer to Problem 17RE

The value of $(fg)(x)$ is $−3x2+5x+2$ .

Domain of $(fg)(x)$ is set of all real numbers.

Explanation of Solution

Given information:

The functions are $f(x)=2−x$ and $g(x)=3x+1$ .

Formula Used:

Let f and g be two functions.

Use the product formula, $(fg)(x)=f(x)g(x)$ .

For a composite function $fg$ , the domain of the function is defined by

$(Domain of f)∩(Domain of g)$

If domain is not specified for a function, one should consider the set of largest real number where the function is defined all over.

Calculation:

The functions are $f(x)=2−x$ and $g(x)=3x+1$ .

Use the product formula, $(fg)(x)=f(x)g(x)$ .

Substitute $2−x$ for $f(x)$ and $3x+1$ for $g(x)$ in the product formula.

$(fg)(x)=f(x)g(x)=(2−x)(3x+1)=6x+2−3x2−x=−3x2+5x+2$

Hence, the value of $(fg)(x)$ is $−3x2+5x+2$ .

As domain of $f(x)=2−x$ and $g(x)=3x+1$ is set of real numbers.

So, domain of $(fg)(x)$ is set of all real numbers.

To determine

The value of $(fg)(x)$ . Also, find the domain of the function $(fg)$ .

Expert Solution

Answer to Problem 17RE

The value of $(fg)(x)$ is $(2−x)(3x+1)$ and the domain is all the real numbers $x$ except $x=−13$ .

Explanation of Solution

Given information:

The functions are $f(x)=2−x$ and $g(x)=3x+1$ .

Formula Used:

Let f and g be two functions.

Use the division formula, $(fg)(x)=f(x)g(x)$ .

For a composite function $f−g$ , the domain of the function is defined by

$(Domain of f)∩(Domain of g)$

If domain is not specified for a function, one should consider the set of largest real number where the function is defined all over.

Calculation:

The functions are $f(x)=2−x$ and $g(x)=3x+1$ .

Use the division formula, $(fg)(x)=f(x)g(x)$ .

Substitute $2−x$ for $f(x)$ and $3x+1$ for $g(x)$ in the division formula.

$(fg)(x)=f(x)g(x)=(2−x)(3x+1)$

Find the domain of the function.

$3x+1=0x=−13$

The domain is all the real numbers $x$ except $x=−13$ because the denominator becomes zero at $x=−13$ .

Hence, the value of $(fg)(x)$ is $(2−x)(3x+1)$ and the domain is all the real numbers $x$ except $x=−13$ .

Chapter 2, Problem 16RE

Chapter 2, Problem 18RE

Chapter 2 Solutions

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The value of ( f + g ) ( x ) and the domain.

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The value of (f+g)(x)<m:math><m:mrow><m:mrow><m:mo>(</m:mo><m:mrow><m:mi>f</m:mi><m:mo>+</m:mo><m:mi>g</m:mi></m:mrow><m:mo>)</m:mo></m:mrow><m:mrow><m:mo>(</m:mo><m:mi>x</m:mi><m:mo>)</m:mo></m:mrow></m:mrow></m:math> and the domain.

Answer to Problem 17RE

Explanation of Solution

The value of (f−g)(x)<m:math><m:mrow><m:mrow><m:mo>(</m:mo><m:mrow><m:mi>f</m:mi><m:mo>−</m:mo><m:mi>g</m:mi></m:mrow><m:mo>)</m:mo></m:mrow><m:mrow><m:mo>(</m:mo><m:mi>x</m:mi><m:mo>)</m:mo></m:mrow></m:mrow></m:math> and its domain.

Answer to Problem 17RE

Explanation of Solution

The value of (fg)(x)<m:math><m:mrow><m:mrow><m:mo>(</m:mo><m:mrow><m:mi>f</m:mi><m:mi>g</m:mi></m:mrow><m:mo>)</m:mo></m:mrow><m:mrow><m:mo>(</m:mo><m:mi>x</m:mi><m:mo>)</m:mo></m:mrow></m:mrow></m:math> and its domain.

Answer to Problem 17RE

Explanation of Solution

Answer to Problem 17RE

Explanation of Solution

Chapter 2 Solutions

Additional Math Textbook Solutions

The value of $(f+g)(x)$ and the domain.

The value of $(f−g)(x)$ and its domain.

The value of $(fg)(x)$ and its domain.