Question

For each pun of functions f and g below, find f (g(x)) and g(f(x)).Then, determine whether fand g are inverses of each other.Simplify your answers as much as possible.(Assume that your expressions are defined for all x in the domain of the composition.You do not have to indicate the domain.)

Answer 1

We want to figure out if f(x) and g(x) are inverses of each other.

$f(x)=3x\text{ and }g(x)=\frac{x}{3}$

We have to find f(g(x)) and g(f(x)).

$f(g(x))=3(\frac{x}{3})=x$

And;

$g(f(x))=\frac{(3x)}{3}=x$

Now, since ;

$f(g(x))=g(f(x))$

We can conclude that;

$f(x)\text{ and g(x) are inverses of each other}$

b.

$f(x)=2x+3\text{ and }g(x)=\frac{x-3}{2}$

Let us compute f(g(x)) and g(f(x)) to see if these two functions are inverses of each other.

$\begin{gathered} f(g(x))=2(\frac{x-3}{2})+3=x-3+3 \\ f(g(x))=x \end{gathered}$

And;

$\begin{gathered} g(f(x))=\frac{2x+3-3}{2}=\frac{2x}{2} \\ g(f(x))=x \end{gathered}$

Now, since ;

$f(g(x))=g(f(x))$

We can conclude that;

$f(x)\text{ and g(x) are inverses of each other}$