Step-by-step explanation:
To determine the y-intercept, we need to determine where the function passes through the y-axis, which is where [tex]x = 0[/tex].
According to the rules of the piecewise function, for [tex]x = 0[/tex], we need to use the equation [tex]\frac{x}{3} + 2[/tex]:
[tex]\frac{0}{3} + 2[/tex]
[tex]0 + 2[/tex]
[tex]2[/tex]
Therefore, the y-intercept is [tex](0, 2)[/tex].
To determine the x-intercept(s), we need to figure out where the function passes through the x-axis, or where the function is equal to [tex]0[/tex]. To do this, we need to set each component of the piecewise function and see where they pass through the x-axis, and if the [tex]x[/tex] value is in their set:
[tex]\frac{x}{3} + 2 = 0[/tex]
[tex]\frac{x}{3} = -2[/tex]
[tex]x = -6[/tex]
For this component of the piecewise function, [tex]x = -6[/tex] is in the range of valid [tex]x[/tex] values for the component, so [tex](-6, 0)[/tex] is one of the x-intercepts.
[tex]4x - 2 = 0[/tex]
[tex]4x = 2[/tex]
[tex]x = \frac{2}{4}[/tex]
[tex]x = \frac{1}{2}[/tex]
Since this value is not [tex]\geq 1[/tex], it falls outside the range of valid [tex]x[/tex] values for this component, so the only x-intercept for the function is [tex](-6, 0)[/tex].
