The function f left parenthesis x right parenthesis equals 16 x minus 4 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 Superscript negative 1.
​(c) Graph​ f, f Superscript negative 1​, and yequalsx on the same coordinate axes.

### (a) Find the inverse of \( f(x) = 16x - 4 \)

To find the inverse, follow these steps:

1. Start by writing \( y = f(x) = 16x - 4 \).
2. Swap \( x \) and \( y \) to reflect the inverse function:  
   \( x = 16y - 4 \).
3. Solve for \( y \):
   \[
   x + 4 = 16y
   \]
   \[
   y = \frac{x + 4}{16}
   \]
So, the inverse function is \( f^{-1}(x) = \frac{x + 4}{16} \).

### Check the inverse:
To verify, we check if \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \).

- First, \( f(f^{-1}(x)) \):
   \[
   f\left(f^{-1}(x)\right) = f\left(\frac{x + 4}{16}\right) = 16\left(\frac{x + 4}{16}\right) - 4 = x + 4 - 4 = x
   \]
- Second, \( f^{-1}(f(x)) \):
   \[
   f^{-1}(f(x)) = f^{-1}(16x - 4) = \frac{(16x - 4) + 4}{16} = \frac{16x}{16} = x
   \]

Both compositions return \( x \), so the inverse is correct.

### (b) Find the domain and range of \( f \) and \( f^{-1} \)

- **Domain of \( f(x) = 16x - 4 \)**: Since \( f(x) \) is a linear function, the domain is all real numbers:  
  \[
  \text{Domain of } f = (-\infty, \infty)
  \]
- **Range of \( f(x) \)**: As a linear function, \( f(x) \) can take any real value, so the range is:  
  \[
  \text{Range of } f = (-\infty, \infty)
  \]
- **Domain of \( f^{-1}(x) = \frac{x + 4}{16} \)**: The inverse is also linear, so the domain is all real numbers:  
  \[
  \text{Domain of } f^{-1} = (-\infty, \infty)
  \]
- **Range of \( f^{-1}(x) \)**: Since the inverse function is linear, the range is also all real numbers:  
  \[
  \text{Range of } f^{-1} = (-\infty, \infty)
  \]

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

To graph these three functions:
1. **\( f(x) = 16x - 4 \)**: This is a straight line with slope 16 and y-intercept -4.
2. **\( f^{-1}(x) = \frac{x + 4}{16} \)**: This is a straight line with slope \( \frac{1}{16} \) and y-intercept \( \frac{1}{4} \).
3. **\( y = x \)**: This is a diagonal line passing through the origin with a slope of 1.

The graph of \( f(x) \) and \( f^{-1}(x) \) should be symmetric about the line \( y = x \).

Let me know if you'd like me to generate a graph.

### Follow-up Questions:
1. Can you explain why the graph of a function and its inverse are symmetric about the line \( y = x \)?
2. How would you find the inverse of a non-linear function?
3. Why is it important to check both compositions when verifying the inverse function?
4. Could a function that is not one-to-one have an inverse?
5. What is the geometric meaning of the domain and range in a function’s graph?

### Tip:
Always remember that a function must be one-to-one (pass the horizontal line test) to have an inverse that is also a function.

The function f(x) = 16x - 4 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.

Learn how to find the inverse of f(x) = 16x - 4, verify the solution, determine the domain and range of both the function and its inverse, and graph f, f⁻¹, and y=x. Suitable for students in Grades 10-12, this problem covers key algebraic concepts and graphing techniques.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation