Let
f
(
x
)
=
x
+
4
x
+
8
 

f
−
1
(
−
5
)
=

The given function is:

\[
f(x) = x + 4x + 8 = 5x + 8
\]

We are tasked with finding \( f^{-1}(-5) \), which means we need to determine the inverse of \( f(x) \) and then substitute \( -5 \) for \( y \) in the inverse.

### Step 1: Find the inverse of \( f(x) \)
To find the inverse, we swap \( f(x) \) with \( y \), so:

\[
y = 5x + 8
\]

Now, solve for \( x \) in terms of \( y \):

\[
y - 8 = 5x
\]

\[
x = \frac{y - 8}{5}
\]

Thus, the inverse function is:

\[
f^{-1}(y) = \frac{y - 8}{5}
\]

### Step 2: Find \( f^{-1}(-5) \)

Substitute \( -5 \) for \( y \) in the inverse function:

\[
f^{-1}(-5) = \frac{-5 - 8}{5} = \frac{-13}{5}
\]

So, \( f^{-1}(-5) = -\frac{13}{5} \).

Would you like more details or have any questions?

Here are 5 related questions you could explore:
1. How do you verify if a function has an inverse?
2. What are the steps to find the inverse of a more complex function?
3. How do you determine if a function is one-to-one?
4. What is the graphical interpretation of an inverse function?
5. Can all linear functions have an inverse?

**Tip:** When finding the inverse of a function, remember to switch the roles of \( x \) and \( y \) and then solve for \( x \).

Learn how to find the inverse of the linear function f(x) = 5x + 8 and solve for f^(-1)(-5) with a step-by-step explanation. This example covers key algebraic concepts such as inverse functions and linear equations. Suitable for students in Grades 9-11.

Math Problem Statement

Let f ( x )

f − 1 ( − 5 )

Solution

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