Scott baked a casserole at an internal temperature of initially 165 degrees and he set the preheat temperature to 325 degrees.  The oven heats up by 40 degrees per minute.  a. Write function T(x) for the temperature (in degrees) after x minutes.  T(x)=     b. Find the inverse function, T exponential -1 (x)  T-1(x)=     c. Find the domain of T exponential -1 (x) in interval notation    d. Find the range of T exponential -1 (x) in interval notation

We will break down the problem step by step:

### Part a: Writing the function \( T(x) \)

The problem describes the heating process, where the casserole starts at 165°F and the temperature increases by 40°F per minute. Thus, the temperature after \( x \) minutes can be expressed as:

\[
T(x) = 165 + 40x
\]

This function represents the temperature of the casserole after \( x \) minutes.

### Part b: Finding the inverse function \( T^{-1}(x) \)

To find the inverse of the function \( T(x) \), we need to solve for \( x \) in terms of \( T \). Starting from the equation:

\[
T(x) = 165 + 40x
\]

To find \( x \) as a function of \( T \), subtract 165 from both sides:

\[
T - 165 = 40x
\]

Now, divide both sides by 40:

\[
x = \frac{T - 165}{40}
\]

Thus, the inverse function is:

\[
T^{-1}(x) = \frac{x - 165}{40}
\]

### Part c: Domain of \( T^{-1}(x) \)

The domain of \( T^{-1}(x) \) corresponds to the range of \( T(x) \). Since the temperature function \( T(x) = 165 + 40x \) starts at 165°F and increases as \( x \) increases, the domain of \( T^{-1}(x) \) must be:

\[
x \geq 165
\]

In interval notation, the domain is:

\[
[165, \infty)
\]

### Part d: Range of \( T^{-1}(x) \)

The range of \( T^{-1}(x) \) corresponds to the domain of \( T(x) \). Since \( T(x) = 165 + 40x \), and \( x \) can take any non-negative value (since time starts at zero and can continue indefinitely), the range of \( T^{-1}(x) \) is all non-negative numbers:

\[
[0, \infty)
\]

### Summary:

a. \( T(x) = 165 + 40x \)

b. \( T^{-1}(x) = \frac{x - 165}{40} \)

c. Domain of \( T^{-1}(x) \): \( [165, \infty) \)

d. Range of \( T^{-1}(x) \): \( [0, \infty) \)

Would you like more details or have any questions?  

Here are 5 related questions:
1. How does the graph of \( T(x) \) look?
2. How do you interpret the slope of \( T(x) \) in real-life terms?
3. How can you verify that \( T^{-1}(x) \) is truly the inverse of \( T(x) \)?
4. What would change in the equation if the casserole started at a different temperature?
5. What if the oven heated up by 30°F per minute instead?

**Tip**: When finding an inverse function, always check that the original function is one-to-one, meaning it passes the horizontal line test.

Scott baked a casserole at an internal temperature of initially 165 degrees and he set the preheat temperature to 325 degrees. The oven heats up by 40 degrees per minute. a. Write function T(x) for the temperature (in degrees) after x minutes. T(x)= b. Find the inverse function, T exponential -1 (x) T-1(x)= c. Find the domain of T exponential -1 (x) in interval notation d. Find the range of T exponential -1 (x) in interval notation

Solve the problem of a casserole's temperature starting at 165°F, heating at 40°F per minute. Learn how to write the function T(x), find its inverse, and determine the domain and range. Suitable for Grades 9-11.

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