Question

Which pair of functions are inverse functions?()=3+5f(x)=3x+5and()=−3−5g(x)=−3x−5 ()=−+57f(x)=−x+57and()=−7+5g(x)=−7x+5 ()=−3−57f(x)=−3x−57and()=3+57g(x)=3x+57 ()=3−5f(x)=3x−5and()=−53

Answer 1

$\begin{gathered} \mathbf{f(x)=\frac{-x+5}{7}} \\ \mathbf{g(x)}=f^{-1}(x)=\mathbf{-7x+5} \end{gathered}$

1) Let's examine the f(x) functions and find the inverse function of f(x), in the first pair of functions:

a) At first, let's swap x for y in the original function

$\begin{gathered} f(x)=3x+5 \\ y=3x+5 \\ x=3y+5 \\ -3y=-x+5 \\ 3y=\text{ x-5} \\ \frac{3y}{3}=\frac{x-5}{3} \\ y=\frac{x-5}{3}\text{ } \\ f^{-1}(x)=\frac{x-5}{3} \end{gathered}$

Note that after swapping x for y, we can isolate y on the left side. So as regards g(x) this is not the inverse function of f(x)

2) Similarly, let's check for f(x)

$\begin{gathered} f(x)=\frac{-x+5}{7} \\ y=\frac{-x+5}{7} \\ x=\frac{-y+5}{7} \\ 7x=-y+5 \\ y=-7x+5 \\ f^{-1}(x)=-7x+5 \end{gathered}$

Note that in this case, we can state that these are inverse functions

$f^{-1}(x)=g(x)$

3) Finally, let's find out the last pair of functions.

$\begin{gathered} f(x)=\frac{-3x-5}{7} \\ y=\frac{-3x-5}{7} \\ x=\frac{-3y-5}{7} \\ 7x=-3y-5 \\ 3y=-7x-5 \\ f^{-1}(x)=\frac{-7x-5}{3} \end{gathered}$

So in this pair, g(x) is not the inverse function of f(x).

4) Hence, the answer is following pair:

$\begin{gathered} f(x)=\frac{-x+5}{7}\text{ } \\ g(x)=f^{-1}(x)=-7x+5 \end{gathered}$