Let's work through the problem step-by-step.

Given the function $f(x) = 9x + 3$, we are asked to:

(a) Find the inverse of $f$ and check the answer.

To find the inverse of a function, follow these steps:

1. Start with $f(x) = 9x + 3$.
2. Replace $f(x)$ with $y$: $y = 9x + 3$.
3. Solve for $x$ in terms of $y$:
   • Subtract 3 from both sides: $y - 3 = 9x$.
   • Divide by 9: $x = \frac{y - 3}{9}$.
4. Replace $x$ with $f^{-1}(x)$ and $y$ with $x$: $f^{-1}(x) = \frac{x - 3}{9}.$

Thus, the inverse function is: $f^{-1}(x) = \frac{x - 3}{9}.$

(b) Find the domain and range of $f$ and $f^{-1}$.

• The domain of $f(x) = 9x + 3$ is all real numbers $(-\infty, \infty)$, because it is a linear function.
• The range of $f(x) = 9x + 3$ is also all real numbers $(-\infty, \infty)$.

For the inverse:

• The domain of $f^{-1}(x) = \frac{x - 3}{9}$ is all real numbers $(-\infty, \infty)$, as it is also a linear function.
• The range of $f^{-1}(x)$ is all real numbers $(-\infty, \infty)$.

(c) Graph $f$, $f^{-1}$, and $y = x$ on the same coordinate axes.

• The function $f(x) = 9x + 3$ is a straight line with slope 9 and y-intercept 3.
• The function $f^{-1}(x) = \frac{x - 3}{9}$ is a straight line with slope $\frac{1}{9}$ and y-intercept $-\frac{1}{3}$.
• The line $y = x$ is the line of symmetry between the function and its inverse.

To check that $f^{-1}(x)$ is indeed the inverse of $f(x)$, we can verify that:

• $f(f^{-1}(x)) = x$
• $f^{-1}(f(x)) = x$

Would you like a detailed explanation for part (c), including plotting the graphs?

Here are five questions to deepen your understanding:

1. How do you derive the formula for the inverse function in more complex cases?
2. What happens if a function is not one-to-one? Can we still find its inverse?
3. How would you explain the concept of inverse functions to someone unfamiliar with it?
4. What is the geometric significance of the line $y = x$ when comparing a function and its inverse?
5. How would the domain and range change if the original function were restricted?

Tip: Always ensure that a function is one-to-one (passes the horizontal line test) before finding its inverse, as not all functions have inverses.