\(\fbox{Option C}\) 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 https://brainly.com/question/2142762
2. Learn more about range of the functions https://brainly.com/question/3412497
3. Learn more about relation and function https://brainly.com/question/1691598
Answer details:
Grade: High school
Subject: Mathematics
Chapter: Functions
Keywords: functions, range, domain, inverse, reverse, fraction, relation, expression, \(\left( {g \circ f} \right)\left( x \right)\), \(\left( {f \circ g} \right)\left( x \right)\).
