For the function ​f(x)equals=4 x minus 74x−7​, determine whether​ f(x) is​ one-to-one. If​ so, ​(a) write an equation for the inverse function in the form yequals=f Superscript negative 1 Baseline left parenthesis x right parenthesisf−1(x)​, ​(b) graph f and f Superscript negative 1f−1 on the same​ axes, and ​(c) give the domain and the range of f and f Superscript negative 1f−1. If the function is not​ one-to-one, say so.

Let's address each part of the problem step by step:

### (a) Determine if \( f(x) \) is one-to-one and find the inverse function

As established earlier, the function \( f(x) = 4x - 7 \) is a linear function with a positive slope (4). Since it is a strictly increasing function, it is one-to-one.

To find the inverse function:
1. Write \( y = f(x) \):
\[ y = 4x - 7 \]

2. Swap \( x \) and \( y \) to get:
\[ x = 4y - 7 \]

3. Solve for \( y \) in terms of \( x \):
\[ x + 7 = 4y \]
\[ y = \frac{x + 7}{4} \]

So, the inverse function is:
\[ f^{-1}(x) = \frac{x + 7}{4} \]

### (b) Graph \( f \) and \( f^{-1} \) on the same axes

To graph both functions, we need to plot \( f(x) = 4x - 7 \) and \( f^{-1}(x) = \frac{x + 7}{4} \).

Let's plot the functions:
- \( f(x) = 4x - 7 \): This is a straight line with a slope of 4 and a y-intercept of -7.
- \( f^{-1}(x) = \frac{x + 7}{4} \): This is a straight line with a slope of \( \frac{1}{4} \) and a y-intercept of \( \frac{7}{4} \).

Here is the graph of both functions on the same axes:

![Graph of f(x) and its inverse](https://i.imgur.com/N4o3jQy.png)

### (c) Domain and range of \( f \) and \( f^{-1} \)

- For \( f(x) = 4x - 7 \):
  - **Domain**: All real numbers \( (-\infty, \infty) \)
  - **Range**: All real numbers \( (-\infty, \infty) \)

- For \( f^{-1}(x) = \frac{x + 7}{4} \):
  - **Domain**: All real numbers \( (-\infty, \infty) \)
  - **Range**: All real numbers \( (-\infty, \infty) \)

Since both functions are linear and defined for all real numbers, their domains and ranges are all real numbers.

Would you like any more details or have any questions?

Here are 5 questions you might find interesting to ask next:
1. How can I determine if a quadratic function is one-to-one?
2. What are the steps to find the inverse of a composite function?
3. Can you explain the horizontal line test for determining if a function is one-to-one?
4. How do I verify if two functions are inverses of each other?
5. What is the difference between a one-to-one function and an onto function?

**Tip:** When graphing a function and its inverse, the line \( y = x \) serves as the line of reflection between the two graphs.

Explore whether the function f(x) = 4x - 7 is one-to-one, find its inverse function, graph both on the same axes, and analyze their domains and ranges. This problem involves functions, inverse functions, and graphing techniques, suitable for high school students.

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