Interpreting the Inverse of a Tabular Function

A function $f (t)$ is given in Table 3, showing distance in miles that a car has traveled in $t$ minutes. Find and interpret $f^{- 1} (70) .$

$t (minutes)$	30	50	70	90
$f (t) (miles)$	20	40	60	70

Table 3

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
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 $t = 90.$

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Using Table 4, find and interpret ⓐ $f(60),$ and ⓑ $f^{- 1}(60).$

$t (minutes)$	30	50	60	70	90
$f (t) (miles)$	20	40	50	60	70

Table 4

How did it go?