Question

How do you find $\left( f*g \right)\left( x \right)$ and $\left( g*f \right)\left( x \right)$. Also determine if the given functions are inverses of each other $f\left( x \right)={{x}^{2}}-3$ and $g\left( x \right)=\sqrt{x}+3$?

Answer 1

Hint: We first explain the meaning of composite function. We try to use the latter half of the problem as an example to understand the concept better. We also find the composite values of $\left( f*g \right)\left( x \right)$ and $\left( g*f \right)\left( x \right)$ for the given functions to find if those functions are inverse of each other or not.

Complete step-by-step solution:
First, we need to explain on how to find the values of $\left( f*g \right)\left( x \right)$ and $\left( g*f \right)\left( x \right)$.
Here the operation $*$ indicates the composite function.
In case of $\left( f*g \right)\left( x \right)$, we first find the value of the function $g\left( x \right)$. Then we place the value of $g\left( x \right)$ in the place of $x$ in $f\left( x \right)$. This means we are finding the value of $f\left( x \right)$ at $x=g\left( x \right)$.
The same goes for $\left( g*f \right)\left( x \right)$, we first find the value of the function $f\left( x \right)$. Then we place the value of $f\left( x \right)$ in the place of $x$ in $g\left( x \right)$. This means we are finding the value of $g\left( x \right)$ at $x=f\left( x \right)$.
For the second part we need to determine if functions $f\left( x \right)={{x}^{2}}-3$ and $g\left( x \right)=\sqrt{x}+3$ are inverse of each other or not.
This will work as an example to understand the composite function better.
Now if $f\left( x \right)$ and $g\left( x \right)$ are inverse of each other then we have $f\left( x \right)={{\left[ g\left( x \right) \right]}^{-1}}$.
This gives \[g\left[ f\left( x \right) \right]=g\left[ {{\left[ g\left( x \right) \right]}^{-1}} \right]=x\]. We applied a composite function to find the condition.
Now we take \[g\left[ f\left( x \right) \right]\] where $f\left( x \right)={{x}^{2}}-3$ and $g\left( x \right)=\sqrt{x}+3$.
So, \[g\left[ f\left( x \right) \right]=g\left[ {{x}^{2}}-3 \right]=\sqrt{{{x}^{2}}-3}+3\]. Value of \[\sqrt{{{x}^{2}}-3}+3\] is not equal to $x$.
Similarly, we find \[f\left[ g\left( x \right) \right]\] where $f\left( x \right)={{x}^{2}}-3$ and $g\left( x \right)=\sqrt{x}+3$.
So, \[f\left[ g\left( x \right) \right]=f\left[ \sqrt{x}+3 \right]={{\left( \sqrt{x}+3 \right)}^{2}}-3=x+6\sqrt{x}+6\] whose value is not equal to $x$.
Therefore, functions $f\left( x \right)={{x}^{2}}-3$ and $g\left( x \right)=\sqrt{x}+3$ aren’t inverse of each other.

Note: We need to understand the difference between the multiplication of two functions and the composite functions. Composite functions deal with the range and domain of the functions. In some cases, composite functions don’t work, but multiplication works in any case.