3.7 Inverse functions  (Page 4/9)

The inverse function takes an output of $f$ and returns an input for $f.$ So in the expression $f^{-1}(70),$ 70 is an output value of the original function, representing 70 miles. The inverse will return the corresponding input of the original function $f,$ 90 minutes, so $f^{-1}(70)=90.$ The interpretation of this is that, to drive 70 miles, it took 90 minutes.

Alternatively, recall that the definition of the inverse was that if $f(a)=b,$ then $f^{-1}(b)=a.$ By this definition, if we are given $f^{-1}(70)=a,$ then we are looking for a value $a$ so that $f(a)=70.$ In this case, we are looking for a $t$ so that $f(t)=70,$ which is when $t=90.$

To evaluate $g(3),$ we find 3 on the x -axis and find the corresponding output value on the y -axis. The point $\left(3,1\right)$ tells us that $g(3)=1.$

To evaluate $g^{-1}(3),$ recall that by definition $g^{-1}(3)$ means the value of x for which $g(x)=3.$ By looking for the output value 3 on the vertical axis, we find the point $\left(5,3\right)$ on the graph, which means $g(5)=3,$ so by definition, $g^{-1}(3)=5.$ See [link] .