Review of Functions: Learn It 3

Intercepts of a Function

Intercepts are a key feature when graphing and analyzing functions because they provide critical points at which the graph intersects the axes.

The points where the graph of the function intersects the x-axis are known as the x-intercept. The x-intercept indicates where the output f(x) is 0.

These intercepts are also known as the zeros or roots of the function because they satisfy the equation f(x)=0. The x-intercepts determine where the graph of f intersects the x-axis, which gives us more information about the shape of the graph of the function. The graph of a function may never intersect the x-axis, or it may intersect multiple (or even infinitely many) times.

Another point of interest is the y-intercept, if it exists. The y-intercept of a function is the point where the graph of the function crosses the y-axis. It represents the output value when the input value x is 0. In other words, it’s the value of the function f(x) at x=0, given by (0,f(0)).

How to: Given a Function f(x), Find the y– and x-intercepts

Finding the y-intercept:

  1. Plug in zero for the x-value in the function and solve for f(0).
  2. The y-intercept will be at the point (0,f(0)).

Finding the x-intercept:

  1. Set the function equal to zero,f(x)=0, and solve for x to find the roots of the function.
  2. The solutions are the x-intercepts, and they’ll be in the form (x,0), where x represents each root.

Since a function has exactly one output for each input, the graph of a function can have, at most, one y-intercept. If x=0 is in the domain of a function f, then f has exactly one y-intercept. If x=0 is not in the domain of f, then f has no y-intercept.

Consider the function f(x)=4x+2.

  1. Find all zeros of f.
  2. Find the y-intercept (if any).