Question

question 2 of 10
what can you say about the continuous function that generated the following table of values?
a. the function has no x - intercepts.
b. the function has at least one x - intercept.
c. there is not enough information to answer the question.
d. the function has more than one x - intercept.

Explanation:

Step1: Analyze the values of \(y\)

When \(x=-2\), \(y = - 0.625\) (negative value), and when \(x = 0\), \(y=0.5\) (positive value). Since the function is continuous, by the Intermediate Value Theorem, if a function \(y = f(x)\) is continuous on an interval \([a,b]\) and \(f(a)\) and \(f(b)\) have opposite signs (\(f(a)\cdot f(b)<0\)), then there exists at least one \(c\in(a,b)\) such that \(f(c) = 0\). Here \(a=-2\), \(b = 0\), \(f(-2)=-0.625\), \(f(0)=0.5\), and \(-0.625\times0.5=-0.3125<0\).

Step2: Check other intervals (optional for conclusion)

We could also check other intervals, but the existence of one interval \([-2,0]\) where the function changes sign (from negative to positive) is sufficient to conclude the existence of at least one \(x\) - intercept.

Answer:

B. The function has at least one \(x\) - intercept.