Finding the [latex]x[/latex]-intercept of a Line

So far we have only discussed the [latex]y-[/latex]intercepts of functions: the point at which the graph of a function crosses the [latex]y[/latex]-axis. A function may also have an [latex]x[/latex]-intercept, which is the [latex]x[/latex]-coordinate of the point where the graph of a function crosses the [latex]x[/latex]-axis. In other words, it is the input value when the output value is zero.

To find the [latex]x[/latex]-intercept, set the function [latex]f[/latex]([latex]x[/latex]) equal to zero and solve for the value of [latex]x[/latex]. For example, consider the function shown:

[latex]f\left(x\right)=3x - 6[/latex]

Set the function equal to [latex]0[/latex] and solve for [latex]x[/latex].

[latex]\begin{array}{l}0=3x - 6\hfill \\ 6=3x\hfill \\ 2=x\hfill \\ x=2\hfill \end{array}[/latex]

The graph of the function crosses the [latex]x[/latex]-axis at the point [latex](2, 0)[/latex].

Q & A Do all linear functions have [latex]x[/latex]-intercepts? No. However, linear functions of the form [latex]y = c[/latex], where [latex]c[/latex] is a nonzero real number are the only examples of linear functions with no [latex]x[/latex]-intercept. For example, [latex]y = 5[/latex] is a horizontal line [latex]5[/latex] units above the [latex]x[/latex]-axis. This function has no [latex]x[/latex]-intercepts.

Find the [latex]x[/latex]-intercept of [latex]f\left(x\right)=\frac{1}{2}x - 3[/latex].

Show Solution

Set the function equal to zero to solve for [latex]x[/latex].

[latex]\begin{array}{l}0=\frac{1}{2}x - 3\\ 3=\frac{1}{2}x\\ 6=x\\ x=6\end{array}[/latex]

The graph crosses the [latex]x[/latex]-axis at the point [latex](6, 0)[/latex]. Analysis of the Solution A graph of the function is shown below. We can see that the [latex]x[/latex]-intercept is [latex](6, 0)[/latex] as expected.

The graph of the linear function [latex]f\left(x\right)=\frac{1}{2}x - 3[/latex].