To solve this problem, let's go through each part carefully.

Given Information

1. We are given a function: $f(x) = \frac{1 - 5x}{x + 2}$ with the restriction $x \neq -2$.

2. We are asked to find the inverse function $f^{-1}(x)$ and specifically evaluate $f^{-1}(-3)$.

Steps to Find the Inverse Function $f^{-1}(x)$

To find the inverse of $f(x)$, we'll switch $y$ and $x$ and then solve for $y$:

1. Set $y = f(x)$, so: $y = \frac{1 - 5x}{x + 2}$

2. Replace $y$ with $x$ and $x$ with $y$ (this is to find $f^{-1}(x)$): $x = \frac{1 - 5y}{y + 2}$

3. Multiply both sides by $y + 2$ to clear the denominator: $x(y + 2) = 1 - 5y$

4. Expand and rearrange to solve for $y$: $xy + 2x = 1 - 5y$ $xy + 5y = 1 - 2x$ $y(x + 5) = 1 - 2x$ $y = \frac{1 - 2x}{x + 5}$

So, the inverse function is: $f^{-1}(x) = \frac{1 - 2x}{x + 5}$

Finding $f^{-1}(-3)$

Now we substitute $x = -3$ into $f^{-1}(x)$:

$f^{-1}(-3) = \frac{1 - 2(-3)}{-3 + 5}$

1. Simplify the numerator: $1 - 2(-3) = 1 + 6 = 7$

2. Simplify the denominator: $-3 + 5 = 2$

3. So, $f^{-1}(-3) = \frac{7}{2}$

Answer

The correct answer is: $\boxed{\frac{7}{2}}$

This corresponds to option (e).

Would you like a more detailed explanation or have any questions?

Related Questions

1. How do we determine the domain and range of an inverse function?
2. How can we verify if two functions are inverses of each other?
3. What are the restrictions on a function to ensure it has an inverse?
4. How can we find inverses for functions with higher degrees, like quadratics?
5. How does the graphical interpretation of a function and its inverse relate?

Tip

When finding the inverse of a rational function, clear the fraction by multiplying both sides by the denominator first, then isolate terms with $y$ to solve for it.