Problem 5 Finding the Inverse of a Formula

The function d = 4.9 t 2 d equals . 4.9 t squared  represents the distance d, in meters, that an object falls in t seconds due to Earth's gravity. Find the inverse of this function. How long, in seconds, does it take for the cliff diver shown to reach the water below?

d = 4.9 t 2   t 2 = d 4.9 Solve for  t . Do not switch the variables . t = d 4.9 Time must be nonnegative .   = 24 4.9 Substitute  24  for  d .   ≈ 2.2 Use a calculator . table with 5 rows and 3 columns , row1 column 1 , d , column 2 equals 4.9 , t squared , column 3 , row2 column 1 , t squared , column 2 equals , d over 4.9 , column 3 table with 2 rows and 1 column , row1 column 1 , cap solvefor . t . , row2 column 1 , cap donotswitchthevariables . . , end table , row3 column 1 , t , column 2 equals , square root of d over 4.9 end root , column 3 cap timemustbenonnegative . . , row4 column 1 , , column 2 equals , square root of 24 over 4.9 end root , column 3 cap substitute . 24 , for , d . , row5 column 1 , , column 2 almost equal to 2.2 , column 3 cap useacalculator . . , end table

It will take about 2.2 seconds for the diver to reach the water.

✓ Got It?

5. The function d = v 2 19.6 d equals . fraction v squared , over 19.6 end fraction  relates the distance d, in meters, that an object has fallen to its velocity v, in meters per second. Find the inverse of this function. What is the velocity of the cliff diver in meters per second as he enters the water?

Take Note Key Concept: Composition of Inverse Functions

If f and f − 1 f super negative 1 end super  are inverse functions, then

( f − 1 ∘ f ) ( x ) = x  and  ( f ∘ f − 1 ) ( x ) = x open , f super negative 1 end super , composition f close open x close equals x , and , open f composition , f super negative 1 end super , close open x close equals x  for x in the domains of f and f − 1 , f super negative 1 end super , comma  respectively.