Question

How would you describe the relationship between the real zero(s) and x-intercept(s) of the function

When you set the function equal to zero, the solution is x = 1; therefore, the graph has an x-intercept of (1, 0).
When you set the function equal to zero, the solutions are x = 0 or x = 1; therefore, the graph has x-intercepts at (0, 0) and (1, 0).
When you substitute x = 0 into the function, there is no solution; therefore, the graph will not have any x-intercepts.
Since there are asymptotes at x = –3, x = –1, and x = 0, the graph has no x-intercepts and, therefore, no real zeros

Answer 1

A real zero of a function is an x-value a for which f(a) = 0. An x-intercept is the point (a, 0) on the graph. So the basic rule is:

- If x = a is a real solution of f(x) = 0 and f is defined at a, then the graph has an x-intercept at (a, 0).
- If f(x) = 0 has no real solutions, the graph has no x-intercepts.
- If f is undefined at a (vertical asymptote or a removable hole), then even if a appears as a root of a simplified form, the original graph does not include (a,0).

Also useful: multiplicity of a zero matters — odd multiplicity zeros cross the x-axis, even multiplicity zeros touch and turn around.

Apply this to the four statements you listed:
1. Correct, provided f(1) is defined. Then (1,0) is an x-intercept.
2. Correct, provided f(0) and f(1) are defined. Then (0,0) and (1,0) are x-intercepts.
3. If you only substitute x = 0 and find no solution, that only tells you x = 0 is not an x-intercept; it does not rule out x-intercepts elsewhere. If f(x) = 0 has no real solution at all, then there are no x-intercepts.
4. The presence of vertical asymptotes at x = −3, −1, and 0 means the function is undefined there (so those x-values cannot be x-intercepts), but it does not imply the graph has no x-intercepts elsewhere.

Summary: Solutions of f(x) = 0 that lie in the domain correspond exactly to x-intercepts (with behavior determined by multiplicity); values not in the domain (asymptotes or holes) cannot be x-intercepts.