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. \[ T(h)=48.5-2.5 h \] 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, $x=I(h)$ and $h=T^{-1}(x)$. (a) Which statement best describes $T^{-1}(x)$ ? 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. 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. (b) $T^{-1}(x)=\square$ (c) $T^{-1}(33)=\square$
Solution
Step 1 ：The problem is asking for the inverse function of \(T(h)\) and its value at a certain point. The inverse function, \(T^{-1}(x)\), will give us the height above the surface when the temperature is \(x\) degrees Celsius.
Step 2 ：To find the inverse function, we need to switch the roles of \(h\) and \(x\) in the original function and solve for \(h\).
Step 3 ：By doing this, we find that the inverse function is \(T^{-1}(x)=19.4 - 0.4x\).
Step 4 ：Then, we can substitute \(x=33\) into the inverse function to find the corresponding height.
Step 5 ：By substituting, we find that \(T^{-1}(33)=6.2\).
Step 6 ：Final Answer: \(\boxed{(a)\ The\ statement\ that\ best\ describes\ T^{-1}(x)\ is\ "The\ height\ above\ the\ surface\ (in\ kilometers)\ when\ the\ temperature\ is\ x\ degrees\ Celsius."\ (b)\ T^{-1}(x)=19.4 - 0.4x\ (c)\ T^{-1}(33)=6.2}\)
