https://file.aiquickdraw.com/mathbot/file/1729021169015jnlwya4q.jpg

The image you've uploaded contains a set of functions \( f(x) \) on the left side and possible corresponding inverses \( f^{-1}(x) \) on the right side. We need to match each function with its correct inverse. Let’s go through the functions and determine the correct inverses step by step.

1. \( f(x) = 6x + 4 \)  
   To find the inverse, we solve for \( x \):
   \[
   y = 6x + 4
   \]
   Subtract 4 from both sides:
   \[
   y - 4 = 6x
   \]
   Divide by 6:
   \[
   x = \frac{y - 4}{6}
   \]
   Therefore, the inverse is:
   \[
   f^{-1}(x) = \frac{x - 4}{6}
   \]
   This corresponds to option **(b)**.

2. \( f(x) = e^x \)  
   The inverse of the exponential function \( e^x \) is the natural logarithm, \( \ln(x) \), so:
   \[
   f^{-1}(x) = \ln(x)
   \]
   This corresponds to option **(c)**.

3. \( f(x) = x^4 \)  
   The inverse of \( x^4 \) is the fourth root of \( x \), so:
   \[
   f^{-1}(x) = \sqrt[4]{x}
   \]
   This corresponds to option **(d)**.

4. \( f(x) = 4^x \)  
   The inverse of \( 4^x \) is the logarithm base 4, so:
   \[
   f^{-1}(x) = \log_4(x)
   \]
   This corresponds to option **(f)**.

5. \( f(x) = x^6 \)  
   The inverse of \( x^6 \) is the sixth root of \( x \), so:
   \[
   f^{-1}(x) = \sqrt[6]{x}
   \]
   This corresponds to option **(e)**.

6. \( f(x) = \sqrt[4]{x - 6} \)  
   To find the inverse, we first square both sides to remove the fourth root:
   \[
   y^4 = x - 6
   \]
   Add 6 to both sides:
   \[
   x = y^4 + 6
   \]
   Therefore, the inverse is:
   \[
   f^{-1}(x) = x^4 + 6
   \]
   This corresponds to option **(g)**.

7. \( f(x) = 6(x + 4) \)  
   This can be simplified as \( f(x) = 6x + 24 \). Solving for the inverse:
   \[
   y = 6x + 24
   \]
   Subtract 24 from both sides:
   \[
   y - 24 = 6x
   \]
   Divide by 6:
   \[
   x = \frac{y - 24}{6}
   \]
   Therefore, the inverse is:
   \[
   f^{-1}(x) = \frac{x}{6} - 4
   \]
   This corresponds to option **(a)**.

### Summary of matches:
1. \( f(x) = 6x + 4 \) → \( f^{-1}(x) = \frac{x - 4}{6} \) (b)
2. \( f(x) = e^x \) → \( f^{-1}(x) = \ln(x) \) (c)
3. \( f(x) = x^4 \) → \( f^{-1}(x) = \sqrt[4]{x} \) (d)
4. \( f(x) = 4^x \) → \( f^{-1}(x) = \log_4(x) \) (f)
5. \( f(x) = x^6 \) → \( f^{-1}(x) = \sqrt[6]{x} \) (e)
6. \( f(x) = \sqrt[4]{x - 6} \) → \( f^{-1}(x) = x^4 + 6 \) (g)
7. \( f(x) = 6(x + 4) \) → \( f^{-1}(x) = \frac{x}{6} - 4 \) (a)

Would you like more details or have any questions about this?

Here are 5 related questions to consider:
1. How do you find the inverse of a function in general?
2. What is the relationship between logarithmic and exponential functions?
3. Why does the function \( f(x) = x^n \) have an inverse only for certain values of \( n \)?
4. Can you explain the process of solving for the inverse step-by-step for different types of functions?
5. How do you determine the domain and range of an inverse function?

**Tip:** When finding the inverse of a function, always remember that you are essentially "undoing" the operations of the original function, reversing each step in the reverse order.

Match each function f(x) with its correct inverse from the given options.

Learn how to match functions with their inverses, including linear, exponential, and power functions. This guide covers essential algebraic and logarithmic concepts, with a step-by-step solution suitable for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation