Question

1. which of the following statements best describes $y = -2x + 5$?

• the function $y = -2x + 5$ is not invertible because it does not pass the horizontal line test.
• the function $y = -2x + 5$ is invertible because it passes both the vertical and horizontal line tests.
• the function $y = -2x + 5$ is invertible because it passes the vertical line test but does not pass horizontal line test.
• the function $y = -2x + 5$ is not invertible because it does not pass the vertical line test.

Explanation:

Step1: Recall Line Test Rules

A function passes the vertical line test (is a function) if no vertical line intersects its graph more than once. A function is invertible (has an inverse function) if it is one - to - one, which means it passes the horizontal line test (no horizontal line intersects its graph more than once).

Step2: Analyze \(y = - 2x+5\)

The equation \(y=-2x + 5\) is a linear function with a slope \(m=-2\) and \(y\) - intercept \(b = 5\). The graph of a linear function \(y=mx + b\) (where \(m
eq0\)) is a straight line with a non - zero slope.

• For the vertical line test: Since it is a linear function (a straight line with non - zero slope), any vertical line will intersect the graph at most once. So it passes the vertical line test (it is a function).
• For the horizontal line test: Since the slope is non - zero, the line is not horizontal. Any horizontal line will intersect the graph of \(y=-2x + 5\) at most once. So it passes the horizontal line test (it is one - to - one).

A function is invertible if and only if it is one - to - one (passes the horizontal line test) and it is a function (passes the vertical line test). So \(y=-2x + 5\) is invertible because it passes both the vertical and horizontal line tests.

Answer:

The function \(y = - 2x+5\) is invertible because it passes both the vertical and horizontal line tests.