https://file.aiquickdraw.com/mathbot/file/17266168398765mgsctny.jpg

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.

The function f(x) = 9x + 3 is one-to-one. (a) Find the inverse of f and check the answer. (b) Find the domain and the range of f and f^{-1}. (c) Graph f, f^{-1}, and y = x on the same coordinate axes.

Explore how to find the inverse of the linear function f(x) = 9x + 3, determine the domain and range for both the function and its inverse, and graph these on the coordinate axes. This tutorial is suitable for high school students in grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation