91Ó°ÊÓ

An inverse function essentially undoes the action of the original function. This means if you start with a value, apply the function, and then apply the inverse function, you end up back at your starting value. For the function given in the exercise, \(f(x) = 5x + 8\), you find the inverse by solving for \(x\) in terms of \(y\). Here's how:
- Begin with the function written in terms of \(y\): \(y = 5x + 8\).
- Isolate \(x\) by performing algebraic operations: first subtract 8 from both sides, \(y - 8 = 5x\), then divide by 5, \(x = \frac{y - 8}{5}\).
- Finally, swap \(x\) and \(y\) to write the inverse function: \( f^{-1}(x) = \frac{x - 8}{5} \).
In simpler terms, an inverse function reverses what the original function does. It allows you to work backwards from a result back to the original input.

The Horizontal Line Test helps to determine if a function is one-to-one. A function is one-to-one if each output (or \(y\)-value) is produced by exactly one input (or \(x\)-value). Here's how you perform the test:
- Imagine drawing horizontal lines (parallel to the \(x\)-axis) through different points on the function's graph.
- Check to see if any horizontal line touches the graph at more than one point.
- If any horizontal line intersects the graph at more than one point, the function is not one-to-one.
In the exercise, the function \(f(x) = 5x + 8\) is a linear function with a constant slope. This means any horizontal line would only intersect the graph at one point. So, the function passes the Horizontal Line Test and is therefore one-to-one.

A linear equation, like \(f(x) = 5x + 8\), represents a straight-line graph. Key aspects of linear equations include:
- The general form is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
- In our function, \(m = 5\), indicating a slope that rises 5 units for every 1 unit it moves to the right.
- The \(y\)-intercept is 8, meaning the line crosses the \(y\)-axis at \(y = 8\).
Linear equations are straightforward and perfect for understanding concepts like one-to-one functions and inverses since each \(x\) maps to exactly one \(y\). This simplicity makes them ideal for beginning algebra students.