Chapter 2.2, Problem 37AYU

To determine

The number of x -intercepts and y -intercepts a graph can have.

Explanation of Solution

The vertical line test states that if we draw a vertical line on the graph of a function, and it cuts it at two or more points, then it is not a function.

The graph of a function can have any number of $x$ intercepts. The number of $x$ intercepts does not violate the vertical line test.

Suppose the graph of $y=f\left(x\right)$ has two $y$ intercepts. Then we can find two points

  $\left(0,a\right)$ and $\left(0,b\right)$ on the graph of $y=f\left(x\right)$ .

For $x=0$ , two values of $y$ result. So two different outputs result from the same input. For this reason, the graph of an equation that is a function can have at most one $y$ intercept.