What is the inverse of the function 
 
\[g(x)=\dfrac{-x-2}{x+4}\] ?
\[g^{-1}(x) =\] 


To find the inverse of the function \( g(x) = \dfrac{-x-2}{x+4} \), we need to follow these steps:

### Step 1: Swap \( g(x) \) with \( y \):
Let \( y = g(x) = \dfrac{-x-2}{x+4} \).

Now, swap \( y \) and \( x \), since finding the inverse involves swapping the roles of \( x \) and \( y \):
\[
x = \dfrac{-y-2}{y+4}
\]

### Step 2: Solve for \( y \):
Now, solve this equation for \( y \).

1. Multiply both sides by \( y + 4 \):
\[
x(y + 4) = -y - 2
\]

2. Distribute the \( x \) on the left-hand side:
\[
xy + 4x = -y - 2
\]

3. Move all terms involving \( y \) to one side of the equation and the other terms to the other side:
\[
xy + y = -4x - 2
\]

4. Factor out \( y \) on the left-hand side:
\[
y(x + 1) = -4x - 2
\]

5. Solve for \( y \):
\[
y = \dfrac{-4x - 2}{x + 1}
\]

Thus, the inverse function is:
\[
g^{-1}(x) = \dfrac{-4x - 2}{x + 1}
\]

Would you like further clarification or a deeper breakdown of the steps?

Here are some related questions to explore further:
1. How can you verify that two functions are inverses of each other?
2. How do asymptotes behave when finding the inverse of a rational function?
3. What is the domain and range of \( g^{-1}(x) \)?
4. How does solving for the inverse change when the function is more complex (e.g., polynomials)?
5. What is the graphical relationship between a function and its inverse?

**Tip:** Always check that \( g(g^{-1}(x)) = x \) and \( g^{-1}(g(x)) = x \) to verify that the inverse is correct!

Learn how to find the inverse function of g(x) = (-x - 2) / (x + 4) step-by-step with detailed explanations and calculations. This problem involves inverse functions and rational functions, suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation