**If f left parenthesis x right parenthesis equals fraction numerator x plus 6 over denominator 5 end fraction, then f to the power of short dash 1 end exponent left parenthesis x right parenthesis = ________.**

To find the inverse function \( f^{-1}(x) \) of the function \( f(x) = \frac{x + 6}{5} \), follow these steps:

1. **Start by setting** \( y = f(x) \):
   \[
   y = \frac{x + 6}{5}
   \]

2. **Swap** \( x \) and \( y \) to find the inverse:
   \[
   x = \frac{y + 6}{5}
   \]

3. **Solve for** \( y \):
   \[
   x \times 5 = y + 6 \quad \text{(Multiply both sides by 5)}
   \]
   \[
   5x = y + 6
   \]
   \[
   y = 5x - 6
   \]

So, the inverse function is:
\[
f^{-1}(x) = 5x - 6
\]

Do you want more details or have any questions?

---

Here are five related questions:
1. How do you verify if two functions are inverses of each other?
2. What is the graphical interpretation of an inverse function?
3. How do you find the inverse of a function that is not linear?
4. Can all functions have an inverse? Why or why not?
5. How does the domain and range of a function relate to its inverse?

**Tip:** To verify that two functions are inverses, compose them (i.e., \( f(f^{-1}(x)) \) and \( f^{-1}(f(x)) \)) and check if both simplify to \( x \).

Discover how to find the inverse function of f(x) = (x + 6) / 5 step-by-step. This problem covers inverse functions in algebra, essential for high school students. Learn to swap variables, solve equations, and determine the inverse function.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation