Q 39. - Precalculus Enhanced with Graphing Utilities

Question

How many x-intercepts can the graph of a function have? How many y-intercepts can the graph of a function have?

Step-by-Step Solution

Answer

Graph of the function has at most one y -intercept and x intercepts are none, any number or infinitely.

Step 1. given information.

We have to check how many x and y intercepts can the graph of functions have.

Step 2. Concept used.

If all vertical lines intersect a curve at most once then the curve represents a function. 

An x-intercept of a graph is a point where the graph crosses the x-axis. 

An x-intercept occurs at a point on the graph where the y-coordinate is zero, (a,0). 

A y-intercept of a graph is a point where the graph crosses the y-axis. 

A y-intercept occurs at a point on the graph where the x-coordinate is zero, (0,b). 

Step 3. Explanation.

A graph can have many intercepts, one intercept, or no intercepts. 

If the graph is the graph of a function then it has at most one y-intercept, 

Since the graph of a function must pass the vertical line test and the y-axis is one such vertical line. 

The graph of a function can have any number of x-intercepts, including none and infinitely many. 

Step 4. Conclusion.

Hence, graph of the function has at most one y -intercept and x intercepts are none, any number or infinitely.