What is the in verse function for f(x)=1/(x+3)

To find the inverse of the function \( f(x) = \frac{1}{x + 3} \), follow these steps:

### Step 1: Replace \( f(x) \) with \( y \)
\[
y = \frac{1}{x + 3}
\]

### Step 2: Swap \( x \) and \( y \)
\[
x = \frac{1}{y + 3}
\]

### Step 3: Solve for \( y \)

Multiply both sides by \( y + 3 \):
\[
x(y + 3) = 1
\]

Distribute \( x \):
\[
xy + 3x = 1
\]

Isolate \( y \):
\[
xy = 1 - 3x
\]

Finally, solve for \( y \) by dividing by \( x \):
\[
y = \frac{1 - 3x}{x}
\]

### Step 4: Rewrite the inverse function

So, the inverse of \( f(x) = \frac{1}{x + 3} \) is:
\[
f^{-1}(x) = \frac{1 - 3x}{x}
\]

Would you like any further details on this, or have any questions? Here are 5 questions that expand on the concept:

1. What are the domain and range of \( f(x) \) and its inverse \( f^{-1}(x) \)?
2. How do we verify if two functions are inverses of each other?
3. What happens to the graph of \( f(x) \) when we graph its inverse?
4. Can inverse functions be found for all types of functions?
5. How do restrictions on the domain affect the inverse function?

**Tip:** Always ensure the original function is one-to-one before attempting to find its inverse.

What is the inverse function for f(x) = 1/(x + 3)?

Learn how to find the inverse of the function f(x) = 1/(x + 3) with this step-by-step solution. Explore key concepts like inverse functions, solving algebraic equations, and the Inverse Function Theorem. This problem is suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation