Find $f(y^2)$ given $f(x)$

Question

This seems sort of like an inverse function but I'm not sure how to find the final answer. Do I just find the inverse of $F(x)$ and then square it?

$F(x) = \frac{1}{x-8}$.

Simplify $F(y^2)$

Answer 1

A function is like a machine that takes an input and gives an output. The specific rules that the machine follows in order to determine the output used are presented in various ways, one of the options being how it is presented here.

$F(\color{red}x)=\dfrac{1}{\color{red}x-8}$

This means, that to get our output, we take our input (in this case $x$), and then subtract eight from it and divide 1 by the result.

We now know how to find, say, $F(\color{red}2)$ as being $F(\color{red}2)=\dfrac{1}{\color{red}2-8}$

This is the same regardless of how complicated or simple of an input we desire, be it a complicated number or even an algebraic expression.

$F(\color{red}{5.88876771}) = \dfrac{1}{\color{red}{5.88876771}-8}$

$F(\color{red}{x^2+5}) = \dfrac{1}{\color{red}{x^2+5}-8}$

Or in your case, even using a variable name different than $x$:

$F(\color{red}{y^2}) = \dfrac{1}{\color{red}{y^2}-8}$

Answer 2

That should be pretty straight forward. To go from f(x) to $f(y^2)$, replace every "x" by "$y^2$". Here, $f(x)= \frac{1}{x- 8}$ so $f(y^2)= \frac{1}{y^2- 8}$.