Show all the work for this problem. determine whether this function has an inverse function. 
1.) y = 3/x-2 - 1
2.)y = x^3 -4x +6
3.) y = ln x^2




Let's analyze each function to determine whether it has an inverse. For a function to have an inverse, it must be **one-to-one**. This means that each value of \( y \) is associated with exactly one value of \( x \). We'll use the **horizontal line test** and the **derivative test** where applicable.

### 1. Function: \( y = \frac{3}{x - 2} - 1 \)

**Step 1: Analyze the Function**
- This function is a transformation of the basic rational function \( y = \frac{1}{x} \).
- The function \( y = \frac{3}{x - 2} \) is a horizontal stretch by a factor of 3 and a horizontal shift to the right by 2 units.
- The subtraction of 1 indicates a downward shift by 1 unit.

**Step 2: Determine if the Function is One-to-One**
- To check if the function is one-to-one, let's rewrite the equation in terms of \( x \) and solve for \( x \).

Starting with:
\[
y = \frac{3}{x - 2} - 1
\]

Add 1 to both sides:
\[
y + 1 = \frac{3}{x - 2}
\]

Now multiply both sides by \( (x - 2) \):
\[
(y + 1)(x - 2) = 3
\]

Distribute \( y + 1 \):
\[
yx + y - 2y - 2 = 3
\]

Solve for \( x \):
\[
x = \frac{3 + 2y}{y + 1}
\]

Since we can solve for \( x \) uniquely in terms of \( y \), the function is **one-to-one**.

**Conclusion:** The function \( y = \frac{3}{x - 2} - 1 \) **has an inverse function**.

### 2. Function: \( y = x^3 - 4x + 6 \)

**Step 1: Analyze the Function**
- The function is a cubic polynomial, which is generally continuous and smooth.

**Step 2: Determine if the Function is One-to-One**
- Let's check the derivative to determine whether the function is strictly increasing or decreasing.

The derivative is:
\[
\frac{dy}{dx} = 3x^2 - 4
\]

**Step 3: Analyze the Derivative**
- Set the derivative equal to zero to find the critical points:
\[
3x^2 - 4 = 0
\]

Solving for \( x \):
\[
x^2 = \frac{4}{3} \quad \Rightarrow \quad x = \pm \frac{2}{\sqrt{3}}
\]

- The function changes its slope at these points, which means the function is not strictly increasing or decreasing over its entire domain.

- Since the function can have the same \( y \) value for different \( x \) values, it **fails the horizontal line test**.

**Conclusion:** The function \( y = x^3 - 4x + 6 \) **does not have an inverse function**.

### 3. Function: \( y = \ln(x^2) \)

**Step 1: Analyze the Function**
- This function involves the natural logarithm of \( x^2 \).
- Let's rewrite the function:
\[
y = \ln(x^2) = 2\ln|x|
\]

**Step 2: Determine if the Function is One-to-One**
- Since \( y = 2\ln|x| \), the function is not one-to-one because \( \ln|x| \) takes the same value for both positive and negative \( x \).

- For example, \( x = 2 \) and \( x = -2 \) give the same \( y \) value, which means it **fails the horizontal line test**.

**Conclusion:** The function \( y = \ln(x^2) \) **does not have an inverse function**.

---

Would you like further details or have any other questions about these problems?

Here are 5 related questions to explore:
1. How does the derivative help determine if a function is one-to-one?
2. What is the general form of the inverse of a linear function?
3. Can a function be one-to-one on a restricted domain even if it’s not one-to-one overall?
4. How does the horizontal line test work in determining the existence of an inverse?
5. What are the characteristics of logarithmic functions and their inverses?

**Tip:** Always consider the domain of the function when determining if it is one-to-one; different domains can change whether a function has an inverse!

Explore whether three mathematical functions have inverse functions through detailed analysis. Includes examples of rational functions, cubic polynomials, and logarithmic functions. Suitable for advanced high school students studying functions and calculus.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation