To solve for the inverse of the function $f(x) = 4x + 6$, let’s follow these steps:

Step 1: Replace $f(x)$ with $y$

We rewrite the function as: $y = 4x + 6$

Step 2: Swap $x$ and $y$

To find the inverse, we swap $x$ and $y$: $x = 4y + 6$

Step 3: Solve for $y$

Isolate $y$ to express it as a function of $x$: $x - 6 = 4y$ $y = \frac{x - 6}{4}$

Thus, the inverse function is: $f^{-1}(x) = \frac{x - 6}{4}$

Step 4: Simplify further

We can rewrite $f^{-1}(x)$ as: $f^{-1}(x) = \frac{1}{4}x - \frac{6}{4}$ $f^{-1}(x) = \frac{1}{4}x - \frac{3}{2}$

Step 5: Match with the given options

The inverse function is: $f^{-1}(x) = \frac{1}{4}x - \frac{3}{2}$

Thus:

• The first blank is $\frac{1}{4}$
• The second blank is $-\frac{3}{2}$

Would you like further clarification or a breakdown of any specific step?

Related Questions:

1. What is the definition of an inverse function and why is it useful?
2. How can we verify if two functions are inverses of each other?
3. What happens to the graph of $f(x) = 4x + 6$ when we graph its inverse?
4. How do we find the inverse for nonlinear functions like $f(x) = x^2 + 2$?
5. How does the slope of a linear function relate to the slope of its inverse?

Tip:

When finding the inverse of a function, always ensure that the original function is one-to-one (passes the Horizontal Line Test), so its inverse is valid.