Find a formula for f Superscript negative 1f−1​(x). Give the domain of f Superscript negative 1f−1​, including any restrictions​ "inherited" from f.
​f(x)equals=44xminus−33

To find the formula for the inverse function \( f^{-1}(x) \) of the function \( f(x) = 4x - 3 \), we need to follow these steps:

### 1. Express the function \( f(x) \):
\[ f(x) = 4x - 3 \]

### 2. Replace \( f(x) \) with \( y \):
\[ y = 4x - 3 \]

### 3. Swap \( x \) and \( y \) to find the inverse:
\[ x = 4y - 3 \]

### 4. Solve for \( y \) to find \( f^{-1}(x) \):
\[ x + 3 = 4y \]
\[ y = \frac{x + 3}{4} \]

So, the inverse function is:
\[ f^{-1}(x) = \frac{x + 3}{4} \]

### 5. Determine the domain of \( f^{-1}(x) \):
Since \( f(x) = 4x - 3 \) is a linear function, it is defined for all real numbers. Therefore, \( f^{-1}(x) \) is also defined for all real numbers. There are no additional restrictions on the domain of \( f^{-1}(x) \) inherited from \( f(x) \).

### Final Answer:
- The inverse function is \( f^{-1}(x) = \frac{x + 3}{4} \).
- The domain of \( f^{-1}(x) \) is all real numbers \( \mathbb{R} \).

Would you like more details or have any questions?

### Related Questions:
1. What are the properties of inverse functions in general?
2. How do you verify that \( f(x) \) and \( f^{-1}(x) \) are truly inverses?
3. Can you find the inverse of a quadratic function, and how does its domain differ?
4. How does the graph of \( f(x) \) relate to the graph of \( f^{-1}(x) \)?
5. What happens to the domain and range when finding the inverse of more complex functions?

### Tip:
Always check that \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \) to verify your inverse function is correct.

Learn how to find the inverse function of f(x) = 4x - 3 with a detailed step-by-step solution. Understand the concept of inverse functions and the formula used to find them. Suitable for 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