A function f ( t ) = h gives the height in feet of a hot air balloon as it ascends over time, in minutes. Find and interpret f^− 1 ( 1500 )

Question

f^-1(1500)=

Answer 1

If we look at the ratio of change in height over time we can get a rough idea of what kind of function f is.

When t=10, f(10) = 500, and when t=20, f(20) = 1000, so then

(1000 - 500) / (20 - 10) = 500 / 10 = 50

We can recognize f(t) to be a linear function where for every t seconds the balloon goes up 50*t feet, so then f(t) = 50t feet

and indeed, checking f(30) = 50(30) = 1500, as the table tells us.

Since f takes a time t as input and outputs a height h, then the inverse f^-1 takes in a height h and outputs a time t, so we want to find f^-1(h) = t . Since f(t) = 50t then we want f^-1(f(t)) = f^-1(50t) = t

Dividing 50t by 50 gives us t, so given a height h=50t we define f^-1(h) = h/50 seconds.

Finishing off the question, given the height 1500 feet we then have f^-1(1500) = (1500)/50 = 30 seconds

Answer 2

Since we are looking for f^-1(x) that means we are inputting (plugging in) a height, and getting a time as an output. from the graph we are finding the y-value, and stating the corresponding x-value. In this example when the height is 1500 feet, the time is 30 seconds.