Example  ELF.INV.2:    (i) Suppose $f(x)=Ab^x$. [$A \ne0$]  Verify that $g(x)=log_b(\frac xA)$ is the inverse function for $f$.
ii) Suppose $f(x)=b^x+k$. Verify that $g(x)=log_b(x−k)$  is the inverse function for $f$.

After checking with the algebra, we check visually with mapping diagrams and graphs.
Which do you find more convincing visually?

(i) $(g \circ f)(x) = g(f(x)) =g(Ab^x) = \log_b ( \frac {Ab^x} A) = \log_b(b^x) = x$.
$(f \circ g)(x) = f(g(x)) =f( \log_b ( \frac xA) = A*b^{\log_b ( \frac xA )}= A* \frac xA= x$.

In the graph frame, click on the play arrow to animate or stop value of $x$ .
Select the point $x$ in the mapping diagram to move the point by mouse or use the scroll up or down key to move by increments.
Use the check boxes in the mapping diagram frame to show or hide the compositions $g\circ f$ or $f \circ g$.
Use the sliders to adjust the values of $b$ and $A$.

(ii) $(g \circ f)(x) = g(f(x)) =g(b^x +k) = \log_b ( (b^x+k)-k) = \log_b(b^x) = x$.
$(f \circ g)(x) = f(g(x)) =f( \log_b ( x-k)) = b^{\log_b ( x-k )}+k= (x-k)+k= x$.

In the graph frame, click on the play arrow to animate or stop value of $x$ .
Select the point $x$ in the mapping diagram to move the point by mouse or use the scroll up or down key to move by increments.
Use the check boxes in the mapping diagram frame to show or hide the compositions $g\circ f$ or $f \circ g$.
Use the sliders to adjust the values of $b$ and $k$.