Question

Bare with me, fractions.

Answer 1

$\fbox{Option C}$ is correct as $\dfrac{1}{3}\left( {3x} \right)$ is equal to $x$ an it is equal to inverse $x$.

Further explanation:

A function that is a reverse of another function is known as an inverse function. If we substitute $x$ in a function $f$ and it gives a result of $y$ then its inverse $z$ to $y$ gives the result $x$.

Given:

The function $f\left( x \right) = 3x$ and $g\left( x \right) = \dfrac{1}{3}x$

We have to verify that $g\left( x \right)$ is the inverse of $f\left( x \right)$.

To verify that $g\left( x \right)$ is the inverse of $f\left( x \right)$ we have to show that $\left( {f \circ g} \right)\left( x \right) = \left( {g \circ f} \right)\left( x \right)$.

First we have to find $\left( {f \circ g} \right)\left( x \right)$.

$\left( {f \circ g}\right)\left( x \right)$ can be obtained as,  

$\left( {f\circ g} \right)\left( x \right) = f\left( {g\left( x \right)} \right)$.

Substitute $\dfrac{1}{3}x$ for $g\left( x \right)$ in above equation to obtain $\left( {f \circ g} \right)\left( x \right)$.

$\begin{gathered}\left( {f \circ g}\right)\left( x \right) = f\left( {\frac{1}{3}x} \right) \\= 3\left( {\frac{1}{3}x}\right)\\=x\end{gathered}$

Now, we have to find $\left( {g \circ f} \right)\left( x \right)$.

$\left( {g \circ f} \right)\left( x \right)$ can be obtained as,  

$\left( {g \circ f} \right)\left( x \right) = g\left( {f\left( x \right)} \right)$.

Substitute $3x$ for $f\left(x\right)$ in above equation to obtain $\left( {g \circ f} \right)\left( x \right)$.

$\begin{gathered}\left( {g \circ f}\right)\left( x \right)= g\left( {3x} \right) \\=\frac{1}{3}\left( {3x}\right)\\=x \\ \end{gathered}$

Here, $\left( {g \circ f} \right)\left( x \right)$ is equal to $x$ and $\left( {f \circ g} \right)\left( x \right)$ is equal to $x$.

Option A is not correct as $3x\left( {\dfrac{x}{3}} \right)$ is equal to ${x^2}$ which is not equal to inverse $x$.

Option B is not correct as $\left( {\dfrac{1}{3}x} \right)\left( {3x} \right)$ is equal to ${x^2}$ which is not equal to inverse $x$.

$\fbox{Option C}$ is correct as $\dfrac{1}{3}\left( {3x} \right)$ is equal to $x$ an it is equal to inverse $x$.

Option D is not correct as $\frac{1}{3}\left( {\frac{1}{3}x} \right)$ is equal to $\frac{1}{9}x$ which is not equal to inverse $x$.

Learn more:

1. Learn more about functions brainly.com/question/2142762

2. Learn more about range of the functions brainly.com/question/3412497

3. Learn more about relation and function brainly.com/question/1691598

Answer details:

Grade: High school

Subject: Mathematics

Chapter: Functions

Keywords: functions, range, domain, inverse, reverse, fraction, relation, expression, $\left( {g \circ f} \right)\left( x \right)$, $\left( {f \circ g} \right)\left( x \right)$.

Answer 2

If we want to verify that g ( x ) is the inverse of f ( x ) we have to show that:
( f ° g ) = ( g ° f )
f ( g ( x ) )=  3 · ( 1/3 x ) = x
g ( f ( x ) ) = 1/3 · ( 3 x ) = x
Answer:
C ) 1/3 ( 3 x )