help with inverse

Question

Scientists have found a relationship between the temperature and the height above a distant planet's surface. T(h), given below, is the temperature in Celsius at a height of h

 kilometers above the planet's surface. The relationship is as follows.

T(h)=48.5-2.5h

Complete the following statements.

let T^-1 be the inverse function of T.

take x to be an output of the function T.

that is, x=T(h) and h=T^-1(x).

(a) which statement best describes T^-1 (x)?

1. the ratio of the temperature (in degrees celcius) to the number of kilometers, x.
2. the height above the surface (in kilometers) when the temperature is x degrees
3. the reciprocal of the temperature (in degrees) at a height of x kilometers
4. the temperature (in degrees) at a height of x kilometers

(b) t^-1 (x) =

(c) T^-1 (33) =

Answer 1

(a) T^-1(x) gives the height above the surface (in kilometers) when the temperature is x degrees.

(b) we need to solve the equation x = 48.5 - 2.5 h for h. This is equivalent to 2.5 h = 48.5 - x, and after dividing both sides by 2.5 we get h = 19.4 - 0.4x. So T^-1(x) = 19.4 - 0.4x.

(c) Plug in x = 33 to get T^-1(33) = 19.4 - (0.4)(33) = 6.2