Question

scientists have found a relationship between the temperature and the height above a distant planets surface. t(h), given below, is the temperature in celsius at a height of h kilometers above the planets surface. the relationship is as follows. t(h)=38 - 1.25h 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)? the height above the surface (in kilometers) when the temperature is x degrees celsius. the temperature (in degrees celsius) at a height of x kilometers. the ratio of the temperature (in degrees celsius) to the number of kilometers, x. the reciprocal of the temperature (in degrees celsius) at a height of x kilometers. (b) t^(-1)(x)=(38 - )/1.25 (c) t^(-1)(25)=10.4

Explanation:

Step1: Understand inverse - function concept

The function $T(h)$ gives temperature $T$ as a function of height $h$. The inverse function $T^{-1}(x)$ takes the temperature $x$ as input and gives the height $h$ as output.

Step2: Find the inverse of the function $T(h)$

We start with $T(h)=38 - 1.25h$. Let $x = T(h)$. Then $x=38 - 1.25h$.
Solve for $h$:
First, add $1.25h$ to both sides: $1.25h+x = 38$.
Then subtract $x$ from both sides: $1.25h=38 - x$.
Finally, divide both sides by $1.25$: $h=\frac{38 - x}{1.25}$, so $T^{-1}(x)=\frac{38 - x}{1.25}$.

Step3: Evaluate $T^{-1}(25)$

Substitute $x = 25$ into $T^{-1}(x)$.
$T^{-1}(25)=\frac{38 - 25}{1.25}=\frac{13}{1.25}=10.4$.

Answer:

(a) The height above the surface (in kilometers) when the temperature is $x$ degrees Celsius.
(b) $T^{-1}(x)=\frac{38 - x}{1.25}$
(c) $10.4$