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The question defines a function \( f(x) = \sqrt{x^3 + 30x - 45} \), and a graph of \( y = f(x) \) is provided. You are asked to:

### (a) Sketch the graph of \( y = f^{-1}(x) \) on the same set of axes.
To graph the inverse function, \( f^{-1}(x) \), we reflect the graph of \( f(x) \) over the line \( y = x \). This reflection interchanges the x- and y-coordinates of the points on the graph of \( f(x) \), creating the inverse.

- Find a few key points on the original graph of \( f(x) \) and then reflect them.
- Plot these reflected points to sketch the graph of \( f^{-1}(x) \).

### (b) Solve the equation \( f(x) = f^{-1}(x) \).
To solve \( f(x) = f^{-1}(x) \), recognize that for an inverse and a function to be equal, the points lie on the line \( y = x \). This implies solving the equation:

\[
f(x) = x
\]

For the given function:

\[
\sqrt{x^3 + 30x - 45} = x
\]

Squaring both sides:

\[
x^3 + 30x - 45 = x^2
\]

Rearranging this into a standard form:

\[
x^3 - x^2 + 30x - 45 = 0
\]

This is a cubic equation, and we can solve it using techniques such as factoring, the Rational Root Theorem, or numerical methods.

Let me know if you'd like help solving this equation further, and if you'd like to see the reflection graph in more detail.

### Follow-up Questions:
1. How do we calculate the inverse of a cubic function?
2. What is the significance of the reflection line \( y = x \) in the graph of inverse functions?
3. Can all functions have inverses? Why or why not?
4. What does the shape of the graph suggest about the behavior of \( f^{-1}(x) \)?
5. How can we use the graph to estimate points where \( f(x) = f^{-1}(x) \)?

**Tip:** When solving cubic equations, checking for possible rational roots first can simplify the process.

The function f(x) is defined by f(x) = √(x^3 + 30x - 45) for x ∈ R. The graph of y = f(x) is shown in the diagram. (a) On the same set of axes, sketch the graph of y = f^(-1)(x). (b) Solve the equation f(x) = f^(-1)(x).

This problem explores the inverse of the function f(x) = √(x^3 + 30x - 45). You'll learn how to graph the inverse function by reflecting over the line y = x and solve the equation where f(x) equals its inverse. Suitable for high school students in Grades 10-12.

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