Q. 65 - Precalculus Enhanced with Graphing Utilities

Question

In Problems 63–72, for the given functions f and g, find the following. For parts (a)–(d), also find the domain.

$$\begin{gathered} \left(a\right) (f+g)\left(x\right) \left(b\right) (f-g)\left(x\right) \left(c\right) (f·g) \left(x\right) \left(d\right) \left(\frac{f}{g}\right)\left(x\right) \\ \left(e\right) (f+g)\left(3\right) \left(f\right) (f-g)\left(4\right) \left(g\right) ( f·g)\left(2\right) \left(h\right) \left(\frac{f}{g}\right)\left(1\right) \end{gathered}$$

$f\left(x\right)=x-1; g\left(x\right)=2x^2$

Step-by-Step Solution

Answer

The value of $(f+g)\left(x\right)=2x^2+x-1$ and the domain of $f+g$ is all set of real number.

$\mathrm{The} \mathrm{value} \mathrm{of} (\mathrm{f}-\mathrm{g})\left(\mathrm{x}\right)=-2{\mathrm{x}}^2+\mathrm{x}-1$and the domain of $f-g$ is set of all real numbers.

$\mathrm{The} \mathrm{value} \mathrm{of} (\mathrm{f}·\mathrm{g})\left(\mathrm{x}\right)=2{\mathrm{x}}^3-2{\mathrm{x}}^2$ and the domain of $f·g$ is set of all real numbers.

$\mathrm{The} \mathrm{value} \mathrm{of} \left(\frac{\mathrm{f}}{\mathrm{g}}\right)\left(\mathrm{x}\right)=\frac{\mathrm{x}-1}{2{\mathrm{x}}^2}$and the domain of $\frac{f}{g} \mathrm{is} \left\{\mathrm{x}\right|\mathrm{x}\ne 0\}$

$$\begin{gathered} \mathrm{The} \mathrm{value} \mathrm{of} (\mathrm{f}+\mathrm{g})\left(3\right)=20 \\ \mathrm{The} \mathrm{value} \mathrm{of} (\mathrm{f}-\mathrm{g})\left(4\right)=-29 \\ \mathrm{The} \mathrm{value} \mathrm{of} (\mathrm{f}·\mathrm{g})\left(2\right)=8 \\ \mathrm{The} \mathrm{value} \mathrm{of} \left(\frac{\mathrm{f}}{\mathrm{g}}\right)\left(1\right)=0 \end{gathered}$$

Step 1. Given Information

In the given problems we have to solve the given functions f and g, find the following. For parts (a)–(d), we have to also find the domain.

$$\begin{gathered} \left(a\right) (f+g)\left(x\right) \left(b\right) (f-g)\left(x\right) \left(c\right) (f·g) \left(x\right) \left(d\right) \left(\frac{f}{g}\right)\left(x\right) \\ \left(e\right) (f+g)\left(3\right) \left(f\right) (f-g)\left(4\right) \left(g\right) ( f·g)\left(2\right) \left(h\right) \left(\frac{f}{g}\right)\left(1\right) \end{gathered}$$

The given function is  $f\left(x\right)=x-1; g\left(x\right)=2x^2$

Step 2. The function f tells us to a number subtract 1. The function g tells us two times of a square of number.

Since these operations can be performed on any real number, we conclude that the domain of $f \mathrm{and} \mathrm{g}$ is the set of all real numbers. 

Part (a) Step 1. We have to find the value of ( f + g ) ( x )

We know that $(f+g)\left(x\right)=f\left(x\right)+g\left(x\right)$

Part (a) Step 2. Putting the value of f ( x )   and   g ( x )

$$\begin{gathered} (f+g)\left(x\right)=x-1+2x^2 \\ (f+g)\left(x\right)=2x^2+x-1 \end{gathered}$$

The domain of $f+g$ consists of those numbers x that are in the domains of both $f \mathrm{and} g$. Therefore, the domain of $f+g$ is all set of real number.

Part (b) Step 1. We have to find the value of ( f - g ) ( x ) We know ( f - g ) ( x ) = f ( x ) - g ( x )

Putting the value of $f\left(x\right) \mathrm{and} g\mathit{\left(}x\mathit{\right)}$

$$\begin{gathered} (f-g)\left(x\right)=x-1-2x^2 \\ (f-g)\left(x\right)=-2x^2+x-1 \end{gathered}$$

The domain of $f-g$ consists of those numbers x that are in the domains of both $f \mathrm{and} g$. Therefore, the domain of $f-g$ is all set of real number.

Part (c) Step 1. We have to find the value of ( f · g )   ( x ) We know that ( f · g )   ( x ) = f ( x ) · g ( x )

Putting the value of $f\left(x\right) \mathrm{and} g\mathit{\left(}x\mathit{\right)}$

$(f·g)\left(x\right)=(x-1)·2x^2$

Using the commutative property

$$\begin{gathered} (f·g)\left(x\right)=x·2x^2-1·2x^2 \\ (f·g)\left(x\right)=2x^3-2x^2 \end{gathered}$$

The domain of $f·g$ consists of those numbers x that are in the domains of both $f \mathrm{and} g$. Therefore, the domain of $f·g$ is all set of real number.

Part (d) Step 1. We have to find the value of f g ( x ) We know that f g ( x ) = f ( x ) g ( x )

Putting the value of $f\left(x\right) \mathrm{and} g\mathit{\left(}x\mathit{\right)}$

$\left(\frac{\mathrm{f}}{\mathrm{g}}\right)\left(\mathrm{x}\right)=\frac{\mathrm{x}-1}{2{\mathrm{x}}^2}$

Part (d) Step 2. The domain of f g consists of the numbers for which g ( x ) ≠ 0 and that are in the domains of both f   and   g .

Since $g\left(x\right)\ne 0$ when $2x^2\ne 0$

divide by 2 on both side

$$\begin{gathered} \frac{2}{2}{x}^2\ne \frac{0}{2} \\ {x}^2\ne 0 \\ x\ne 0 \end{gathered}$$

The domain of $\left\{x\right|x\ne 0\}$

Part (e) Step 1. We have to find the value of ( f + g ) ( 3 ) From the part (a) we know the value of ( f + g ) ( x ) = 2 x 2 + x - 1

Putting $x=3$ in the value of $(f+g)\left(x\right)$

$$\begin{gathered} (f+g)\left(3\right)=2{\left(3\right)}^2+3-1 \\ (f+g)\left(3\right)=2·9+3-1 \\ (f+g)\left(3\right)=18+3-1 \\ (f+g)\left(3\right)=20 \end{gathered}$$

Part (f) Step 1. We have to find the value of ( f - g ) ( 4 ) From the part (b) we know the value of ( f - g ) ( x ) = - 2 x 2 + x - 1

Putting $x=4$ in the value of $(f-g)\left(x\right)$

$$\begin{gathered} (f-g)\left(4\right)=-2{\left(4\right)}^2+4-1 \\ (f-g)\left(4\right)=-2\times 16+4-1 \\ (f-g)\left(4\right)=-32+4-1 \\ (f-g)\left(4\right)=-29 \end{gathered}$$

Part (g) Step 1. 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font-family="round_brackets18549f92a457f2409" font-size="16" text-anchor="middle" x="3.5" y="18">(</text><text font-family="aec8956637a99787bd197eacd77acce" font-size="16" font-style="italic" text-anchor="middle" x="8.5" y="18">f</text><text font-family="math1e5c016e17c2c6a729dd4107924" font-size="16" text-anchor="middle" x="18.5" y="18">&#xB7;</text><text font-family="Arial" font-size="16" font-style="italic" text-anchor="middle" x="26.5" y="18">g</text><text font-family="round_brackets18549f92a457f2409" font-size="16" text-anchor="middle" x="34.5" y="18">)</text><text font-family="round_brackets18549f92a457f2409" font-size="16" text-anchor="middle" x="40.5" y="18">(</text><text font-family="Arial" font-size="16" font-style="italic" text-anchor="middle" x="46.5" y="18">x</text><text font-family="round_brackets18549f92a457f2409" font-size="16" text-anchor="middle" x="53.5" y="18">)</text><text font-family="math1e5c016e17c2c6a729dd4107924" font-size="16" text-anchor="middle" x="64.5" y="18">=</text><text font-family="Arial" font-size="16" text-anchor="middle" x="77.5" y="18">2</text><text font-family="Arial" font-size="16" font-style="italic" text-anchor="middle" x="86.5" y="18">x</text><text font-family="Arial" font-size="12" text-anchor="middle" x="94.5" y="11">3</text><text font-family="math1e5c016e17c2c6a729dd4107924" font-size="16" text-anchor="middle" x="105.5" y="18">&#x2212;</text><text font-family="Arial" font-size="16" text-anchor="middle" x="117.5" y="18">2</text><text font-family="Arial" font-size="16" font-style="italic" text-anchor="middle" x="126.5" y="18">x</text><text font-family="Arial" font-size="12" text-anchor="middle" x="134.5" y="11">2</text></svg>" ( f · g ) ( x ) = 2 x 3 - 2 x 2

Putting $x=2$ in the value of $(f·g)\left(x\right)$

$$\begin{gathered} (f·g)\left(2\right)=2{\left(2\right)}^3-2{\left(2\right)}^2(f·g)\left(2\right)=2·8-2·4 \\ (f·g)\left(2\right)=16-8 \\ (f·g)\left(2\right)=8 \end{gathered}$$

Part (h) Step 1. We have to find the value of f g ( 1 ) From the part (d) we know the value of src="data:image/svg+xml;base64,<svg xmlns="http://www.w3.org/2000/svg" xmlns:wrs="http://www.wiris.com/xml/mathml-extension" height="45" width="110" wrs:baseline="27"><!--MathML: <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mfrac><mi>f</mi><mi>g</mi></mfrac></mfenced><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mfrac><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></math>--><defs><style 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font-family="bracketse552f5417ff4680c6b50499" font-size="16" text-anchor="start" x="1.5" y="16">&#x239B;</text><text font-family="brack_sm2882ad605b1e27be87c7468" font-size="16" text-anchor="start" x="1.5" y="21">&#x239C;</text><text font-family="brack_sm2882ad605b1e27be87c7468" font-size="16" text-anchor="start" x="1.5" y="26">&#x239C;</text><text font-family="bracketse552f5417ff4680c6b50499" font-size="16" text-anchor="start" x="1.5" y="40">&#x239D;</text><text font-family="bracketse552f5417ff4680c6b50499" font-size="16" text-anchor="start" x="26.5" y="16">&#x239E;</text><text font-family="brack_sm2882ad605b1e27be87c7468" font-size="16" text-anchor="start" x="26.5" y="21">&#x239F;</text><text font-family="brack_sm2882ad605b1e27be87c7468" font-size="16" text-anchor="start" x="26.5" y="26">&#x239F;</text><text font-family="bracketse552f5417ff4680c6b50499" font-size="16" text-anchor="start" x="26.5" y="40">&#x23A0;</text><line stroke="#000000" stroke-linecap="square" stroke-width="1" x1="9.5" x2="22.5" y1="21.5" y2="21.5"/><text font-family="aec8956637a99787bd197eacd77acce" font-size="16" font-style="italic" text-anchor="middle" x="15.5" y="15">f</text><text font-family="Arial" font-size="16" font-style="italic" text-anchor="middle" x="15.5" y="38">g</text><text font-family="round_brackets18549f92a457f2409" font-size="16" text-anchor="middle" x="35.5" y="27">(</text><text font-family="Arial" font-size="16" font-style="italic" text-anchor="middle" x="41.5" y="27">x</text><text font-family="round_brackets18549f92a457f2409" font-size="16" text-anchor="middle" x="48.5" y="27">)</text><text font-family="math143f4d31b04031e49f5eb18baba" font-size="16" text-anchor="middle" x="59.5" y="27">=</text><line stroke="#000000" stroke-linecap="square" stroke-width="1" x1="70.5" x2="106.5" y1="21.5" y2="21.5"/><text font-family="Arial" font-size="16" font-style="italic" text-anchor="middle" x="76.5" y="16">x</text><text font-family="math143f4d31b04031e49f5eb18baba" font-size="16" text-anchor="middle" x="88.5" y="16">&#x2212;</text><text font-family="Arial" font-size="16" text-anchor="middle" x="100.5" y="16">1</text><text font-family="Arial" font-size="16" text-anchor="middle" x="80.5" y="41">2</text><text font-family="Arial" font-size="16" font-style="italic" text-anchor="middle" x="89.5" y="41">x</text><text font-family="Arial" font-size="12" text-anchor="middle" x="97.5" y="34">2</text></svg>" localid="1646114788207" src="https://studysmarter-mediafiles.s3.amazonaws.com/media/textbook-exercise-images/0c23ff06-f983-4698-9659-a39726b244db.svg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIA4OLDUDE42UZHAIET%2F20220301%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20220301T070702Z&X-Amz-Expires=90000&X-Amz-SignedHeaders=host&X-Amz-Signature=1d995055aff3a444dff5fbf67ddae02c6941989a04faebd54bc3eb6733092229" src="https://studysmarter-mediafiles.s3.amazonaws.com/media/textbook-exercise-images/0c23ff06-f983-4698-9659-a39726b244db.svg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIA4OLDUDE42UZHAIET%2F20220301%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20220301T065045Z&X-Amz-Expires=90000&X-Amz-SignedHeaders=host&X-Amz-Signature=a1f087885fadbef9ae45c73856550b167ca7a719eaf951ad88bc37c9abda396e" f g ( x ) = x - 1 2 x 2

Putting $x=1$ in the value of $\left(\frac{f}{g}\right)\left(x\right)$

$$\begin{gathered} \left(\frac{f}{g}\right)\left(1\right)=\frac{1-1}{2{\left(1\right)}^2} \\ \left(\frac{f}{g}\right)\left(1\right)=\frac{0}{2} \\ \left(\frac{f}{g}\right)\left(1\right)=0 \end{gathered}$$