Question

How do you find the inverse of $f\left( x \right)=3x-5$ and is it a function?

Answer 1

Hint: To find the inverse of the given function $f\left( x \right)=3x-5$, first of all, we will assume $f\left( x \right)$ as y and write y in place of $f\left( x \right)$ in the above function. Then, we are going to arrange this equation in such a way so that we get x in terms of y. After that, we have to check whether the inverse of the function that we have calculated is a function or not by checking the condition that only one value is possible corresponding to each x value.

Complete step-by-step answer:
The function given above which we have to find the inverse of is:
$f\left( x \right)=3x-5$
Now, we are going to write y in place of $f\left( x \right)$ in the above equation and we get,
 $\Rightarrow y=3x-5$
The function in x written on the R.H.S of the above equation is calculated by writing x in terms of y. For that, we are adding 5 on both the sides of the above equation we get,
$\Rightarrow y+5=3x-5+5$
As you can see that 5 written on the R.H.S of the above equation will be cancelled out and we are left with:
$\Rightarrow y+5=3x$
Now, dividing 3 on both the sides of the above equation we get,
$\Rightarrow \dfrac{y+5}{3}=x$
Now, writing x in place of y and ${{f}^{-1}}\left( x \right)$ in place of x in the above equation we get,
${{f}^{-1}}\left( x \right)=\dfrac{x+5}{3}$
Hence, we have found the inverse of the above function as:
${{f}^{-1}}\left( x \right)=\dfrac{x+5}{3}$
Now, to check whether this inverse is a function or not by taking ${{f}^{-1}}\left( x \right)$ as y in the above equation and then plot the following on the graph we get,
$y=\dfrac{x+5}{3}$

In the above graph, you can see that only one y value is possible corresponding to an x value.
So, this means that the inverse which we have calculated above is a function.

Note: The possible mistake that could be possible in the above problem is that calculation mistake when we are converting x in terms of y so be aware while converting x in terms of y.
In the above solution, we have learnt two concepts. First is, to find the inverse of any function and second is, to check whether an expression is a function or not.