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.