Question

what can you say about the continuous function that generated the following table of values?

x	y
-4	-0.906
-2	-0.625
0	0.5
2	5
4	23

a. the function has more than one x-intercept.
b. the function has at least one x-intercept.
c. the function has no x-intercepts.
d. there is not enough information to answer the question

Explanation:

Step1: Analyze function sign changes

Check \( y \)-values: At \( x = -2 \), \( y = -0.625 \) (negative); at \( x = 0 \), \( y = 0.5 \) (positive). Also, at \( x = -4 \), \( y = -0.906 \) (negative); at \( x = 0 \), \( y = 0.5 \) (positive).

Step2: Apply Intermediate Value Theorem

Since the function is continuous, and \( y \) changes sign (from negative to positive) between \( x = -4 \) to \( x = 0 \) (and \( x = -2 \) to \( x = 0 \)), there must be at least one \( x \)-intercept (where \( y = 0 \)) in those intervals. We can't confirm more than one, but at least one exists.

Answer:

B. The function has at least one x - intercept.