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!