Question

How do you write the inverse function for $f\left( x \right)=\dfrac{1}{2}x+4$ ?

Answer 1

Hint: To find the inverse of the given function, first of all name $f\left( x \right)$ as y in the above problem and then we will write x in terms of y. In the above problem, the function given is written in terms of x. After writing the x in terms of y then substitute y as x. And hence, we will get the inverse of the function.

Complete step by step solution:
The function given in the above problem of which we have to find the inverse of is as follows:
$f\left( x \right)=\dfrac{1}{2}x+4$
Let us rename $f\left( x \right)$ as y in the above equation and we get,
$\Rightarrow y=\dfrac{1}{2}x+4$
As you can see that in the above equation, we have given y as a function in x so to find the inverse of the above function, we are going to rearrange the above equation in such a way so that we will get x as a function in y.
Subtracting 4 on both the sides we get,
$\Rightarrow y-4=\dfrac{1}{2}x$
Multiplying 2 on both the sides we get,
$\Rightarrow 2\left( y-4 \right)=\dfrac{1}{2}x\left( 2 \right)$
In the R.H.S of the above equation, 2 will be cancelled out in the numerator and denominator and we get,
$\Rightarrow 2\left( y-4 \right)=x$
Now, multiplying 2 inside the bracket of the L.H.S of the above equation and we get,
$\Rightarrow 2y-8=x$
Now, we are replacing $x$ as ${{f}^{-1}}\left( x \right)$ and y as $x$ in the above equation and we get,
$\Rightarrow 2x-8={{f}^{-1}}\left( x \right)$
Hence, we have found the inverse of the given function and is equal to $2x-8$.

Note: We can check whether the inverse of the function which we have calculated is correct or not by writing x in ${{f}^{-1}}\left( x \right)$ as f(x) and then see whether we are getting x or not.
Multiplying function with its inverse and we get,
$f\left( x \right)=\dfrac{1}{2}x+4$
$\Rightarrow {{f}^{-1}}\left( x \right)=2x-8$
Substituting x as f(x) in the above inverse function we get,
$\begin{align}
  & \Rightarrow {{f}^{-1}}\left( x \right)=2\left( f\left( x \right) \right)-8 \\
 & \Rightarrow {{f}^{-1}}\left( x \right)=2\left( \dfrac{1}{2}x+4 \right)-8 \\
\end{align}$
Multiplying 2 in the bracket of the above equation we get,
$\begin{align}
  & \Rightarrow {{f}^{-1}}\left( x \right)=x+8-8 \\
 & \Rightarrow {{f}^{-1}}\left( x \right)=x+0 \\
 & \Rightarrow {{f}^{-1}}\left( x \right)=x \\
\end{align}$
As we are getting x so the inverse of the function which we have found out is correct.