Question

Some functions f have the property that $f(a+b)=f\left(a\right)+f\left(b\right)$for all real numbers a and b. Which of the following functions have this property?

$$\begin{gathered} \left(a\right)h\left(x\right)=2x\left(b\right)g\left(x\right)=x^2 \\ \left(c\right)F\left(x\right)=5x-2\left(d\right)G\left(x\right)=\frac{1}{x} \end{gathered}$$

Short answer

The function$h\left(x\right)=2x$have a property of$f(a+b)=f\left(a\right)+f\left(b\right)$.

Step-by-step solution

Step 1. Given Information 

Some functions f have the property that $f(a+b)=f\left(a\right)+f\left(b\right)$for all real numbers a and b.

We have to find which of the following functions have this property.

$$\begin{gathered} \left(a\right)h\left(x\right)=2x\left(b\right)g\left(x\right)=x^2 \\ \left(c\right)F\left(x\right)=5x-2\left(d\right)G\left(x\right)=\frac{1}{x} \end{gathered}$$

Part (a) Step 1. To find h(x)=2x have property 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="21" width="145" wrs:baseline="16"><!--MathML: <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo>)</mo><mo>=</mo><mi>f</mi><mo>(</mo><mi>a</mi><mo>)</mo><mo>+</mo><mi>f</mi><mo>(</mo><mi>b</mi><mo>)</mo></math>--><defs><style 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font-family="a9ae65dcc5ba03ef2f0f633c7df668a" font-size="16" font-style="italic" text-anchor="middle" x="3.5" y="16">f</text><text font-family="round_brackets18549f92a457f2409" font-size="16" text-anchor="middle" x="12.5" y="16">(</text><text font-family="a9ae65dcc5ba03ef2f0f633c7df668a" font-size="16" font-style="italic" text-anchor="middle" x="18.5" y="16">a</text><text font-family="math12ed72e0d2d50af08c235c494fe" font-size="16" text-anchor="middle" x="31.5" y="16">+</text><text font-family="Arial" font-size="16" font-style="italic" text-anchor="middle" x="43.5" y="16">b</text><text font-family="round_brackets18549f92a457f2409" font-size="16" text-anchor="middle" x="51.5" y="16">)</text><text font-family="math12ed72e0d2d50af08c235c494fe" font-size="16" text-anchor="middle" x="62.5" y="16">=</text><text font-family="a9ae65dcc5ba03ef2f0f633c7df668a" font-size="16" font-style="italic" text-anchor="middle" x="74.5" y="16">f</text><text font-family="round_brackets18549f92a457f2409" font-size="16" text-anchor="middle" x="83.5" y="16">(</text><text font-family="a9ae65dcc5ba03ef2f0f633c7df668a" font-size="16" font-style="italic" text-anchor="middle" x="89.5" y="16">a</text><text font-family="round_brackets18549f92a457f2409" font-size="16" text-anchor="middle" x="97.5" y="16">)</text><text font-family="math12ed72e0d2d50af08c235c494fe" font-size="16" text-anchor="middle" x="107.5" y="16">+</text><text font-family="a9ae65dcc5ba03ef2f0f633c7df668a" font-size="16" font-style="italic" text-anchor="middle" x="118.5" y="16">f</text><text font-family="round_brackets18549f92a457f2409" font-size="16" text-anchor="middle" x="127.5" y="16">(</text><text font-family="Arial" font-size="16" font-style="italic" text-anchor="middle" x="133.5" y="16">b</text><text font-family="round_brackets18549f92a457f2409" font-size="16" text-anchor="middle" x="141.5" y="16">)</text></svg>" role="math" localid="1646413011142" src="https://studysmarter-mediafiles.s3.amazonaws.com/media/textbook-exercise-images/4ec980f8-bd32-4338-8e38-92eb55f262b8.svg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIA4OLDUDE42UZHAIET%2F20220304%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20220304T171406Z&X-Amz-Expires=90000&X-Amz-SignedHeaders=host&X-Amz-Signature=3e47c8b0c257eea8a3c4c062f6c31b09508eebbcab11857f0e4433bde908a995" f(a+b)=f(a)+f(b) we solve the function.

To find the property we put $x=a+b$in the given function then solving.

$h(a+b)=2(a+b)$

Using commutative property

$h(a+b)=2a+2b$

Part (a) Step 2. The value of given function after solving is h(a+b)=2a+2b.

This function is in the form of $f(a+b)=f\left(a\right)+f\left(b\right)$.

So this given function have that property.

Part (b) Step 1. To find g(x)=x2 have property of f(a+b)=f(a)+f(b) we solve the function.

To find the property we put $x=a+b$in the given function then solving.

$g(a+b)={(a+b)}^2$

Using the formula${(a+b)}^2=a^2+2ab+b^2$

$g(a+b)={a^2+2ab+b}^2$

Part (b) Step 2. The value of given function after solving is .

This function is not in the form of $f(a+b)=f\left(a\right)+f\left(b\right)$.

So this given function does not have that property.

Part (c) Step 1. To find F(x)=5x-2 have property of f(a+b)=f(a)+f(b) we solve the function.

To find the property we put $x=a+b$in the given function then solving.

$F(a+b)=5(a+b)-2$

Using commutative property

$F(a+b)=5a+5b-2$

Part (c) Step 2. The value of given function after solving is .

This function is not in the form of $f(a+b)=f\left(a\right)+f\left(b\right)$.

So this given function does not have that property.

Part (d) Step 1. To find G(x)=1x have property of f(a+b)=f(a)+f(b) we solve the function.

To find the property we put $x=a+b$in the given function then solving.

$G(a+b)=\frac{1}{(a+b)}$

Part (d) Step 2. The value of given function after solving is .

This function is not in the form of $f(a+b)=f\left(a\right)+f\left(b\right)$.

So this given function does not have that property.