Answer:
[tex]f^{-1}(x)=\frac{1}{2}\sqrt{\frac{x}{2}}[/tex]
Step-by-step explanation:
Given:
The function is given as;
[tex]f(x)=8x^2[/tex]
In order to find the inverse, the steps to be followed are:
Step 1: Replace [tex]f(x)[/tex] by [tex]y[/tex]. This gives,
[tex]y=8x^2[/tex]
Step 2: Switch 'y' by 'x' and 'x' by 'y'. This gives,
[tex]x=8y^2[/tex]
Step 3: Solve for 'y'.
Dividing both sides by 8, we get:
[tex]\frac{x}{8}=\frac{8y^2}{8}[/tex]
[tex]\frac{x}{8}=y^2[/tex] or
[tex]y^2=\frac{x}{8}[/tex]
Taking square root on both sides, we get:
[tex]\sqrt{y^2}=\sqrt{\frac{x}{8}}[/tex]
[tex]y=\frac{1}{2}\sqrt{\frac{x}{2}}[/tex]
Now, we replace 'y' by [tex]f^{-1}(x)[/tex].
Therefore, the inverse of the given function is:
[tex]f^{-1}(x)=\frac{1}{2}\sqrt{\frac{x}{2}}[/tex]
