Question

Match the functions with their inverse functions.

Answer 1

Hello!

The answers are:

$f(x)=\frac{5x+3}{2}$

$f(x)=2x-8$

$f(x)=7x-2$

$f(x)=\frac{1}{x+2}$

Why?

To find the inverse function of a function, we need to replace the variable (x for this case) for the function itself (f(x)), where we see "x" we must rewrite with "y" or "f(x) and then, isolate "y" or "f(x).  If we have the inverse function and we want to find the function, we need to reverse the process.

Let's find each function of the given inverse functions in order to match each one with its inverse.

First inverse function,

$f^{-1}(x)=\frac{2x-3}{5}$

Replacing "x" with "f(x)" and "f-1(x)" with "x" we have:

$x=\frac{2f(x)-3}{5}\\\\5x=2f(x)-3\\\\5x+3=2f(x)\\\\f(x)=\frac{5x+3}{2}$

Therefore, the function that matches with the first inverse function is:

$f(x)=\frac{5x+3}{2}$

Second inverse function,

$f^{-1}(x)=\frac{x+8}{2}$

Replacing "x" with "f(x)" and "f-1(x)" with "x" we have:

$x=\frac{f(x)+8}{2}\\\\2x=f(x)+8\\\\2x-8=f(x)\\\\f(x)=2x-8$

Hence, the function that matches with the second inverse function is:

$f(x)=2x-8$

Third inverse function,

$f^{-1}(x)=\frac{x+2}{7}$

Replacing "x" with "f(x)" and "f-1(x)" with "x" we have:

$x=\frac{f(x)+2}{7}\\\\7x=f(x)+2\\\\7x-2=f(x)\\\\f(x)=7x-2$

So, the function that matches with the third inverse function is:

$f(x)=7x-2$

Fourth inverse function,

$f^{-1}(x)=\frac{1-2x}{x}$

Replacing "x" with "f(x)" and "f-1(x)" with "x" we have:

$x=\frac{1-2f(x)}{f(x)}\\\\xf(x)=1-2f(x)\\\\f(x)x+2f(x)=1\\\\f(x)(x+2)=1\\\\f(x)=\frac{1}{x+2}$

So, the function that matches with the fourth inverse function is:

$f(x)=\frac{1}{x+2}$

Have a nice day!