91Ó°ÊÓ

For a function to have an inverse, it must be a one-to-one function. This means that for every output value, there is exactly one corresponding input value.
A function passes the Horizontal Line Test if no horizontal line intersects its graph more than once. This is a quick way to check if a function is one-to-one.
For example, the given function is \( t(x) = \frac{x - 4}{x + 2} \), and to determine if it's one-to-one:
- If we pick different values for \( x \), they must yield different values of \( t(x) \).
- Conversely, for \( t(x) \) to be the same, the values of \( x \) must be the same as well.

Algebraic manipulation involves rearranging equations to isolate a specific variable. This skill is essential for finding inverse functions.
Here’s how we manipulate the given function:
1. Start with the function \( t(x) = \frac{x - 4}{x + 2} \).
2. Replace \( t(x) \) with \( y \), giving us \( y = \frac{x - 4}{x + 2} \).
3. Swap \( x \) and \( y \) to solve for the inverse: \( x = \frac{y - 4}{y + 2} \).
4. Multiply both sides by \( y + 2 \) to clear the denominator: \( x(y + 2) = y - 4 \).
5. Distribute \( x \): \( xy + 2x = y - 4 \).
6. Combine like terms to get: \( xy - y = -4 - 2x \).
7. Factor out \( y \): \( y(x - 1) = -4 - 2x \).
8. Finally, isolate \( y \): \( y = \frac{-4 - 2x}{x - 1} \).

Function notation is a way to represent functions and their inverses concisely.
In function notation, the original function is written as \( t(x) \), where \( t \) represents the function, and \( x \) is the input variable.
For our specific problem, the function \( t(x) \) represents the formula \( \frac{x - 4}{x + 2} \).
When finding the inverse of a function, we use \( t^{-1}(x) \) notation for the inverse.
The inverse function \( t^{-1}(x) \) essentially reverses the effect of \( t(x) \). Hence, if \( t(x) = y \), then \( t^{-1}(y)\) will return \( x \).
Using function notation helps us understand the relationship between inputs and outputs clearly. It also allows us to conduct algebraic manipulations more systematically and confirm our final equation for the inverse function: \( t^{-1}(x) = \frac{-4 - 2x}{x - 1} \).