Question

question 7 of 10
what can you say about the continuous function that generated the following table of values?

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

Explanation:

Step1: Recall the Intermediate Value Theorem

The Intermediate Value Theorem states that if a function \(y = f(x)\) is continuous on a closed interval \([a,b]\), and \(k\) is a number between \(f(a)\) and \(f(b)\), then there exists at least one number \(c\) in the interval \((a,b)\) such that \(f(c)=k\). For \(x\) - intercepts, we are looking for \(c\) such that \(f(c) = 0\).

Step2: Analyze the sign - changes

We know that when \(x=-3\), \(y = f(-3)=4\) (positive) and when \(x = 0\), \(y=f(0)=-3\) (negative). Since the function is continuous, by the Intermediate Value Theorem, there exists at least one \(x_1\in(-3,0)\) such that \(f(x_1)=0\). Also, when \(x = 5\), \(y=f(5) = 0\).

Answer:

B. the function has more than one \(x\) - intercept