Question

the values in the table represent the graph of a continuous function. which interval must contain an x - intercept?

x	y
-3.1	-1.85
-2.7	-0.84
-2.3	-0.09
-1.9	0.44
-1.5	0.79
-1.1	1.00
-0.7	1.10
-0.3	1.13
0.1	1.13

-3.1 < x < -2.7
-2.3 < x < -1.9
-1.5 < x < -1.1
-0.3 < x < 0.1

Explanation:

Step1: Recall Intermediate - Value Theorem

For a continuous function \(y = f(x)\), if \(f(a)\) and \(f(b)\) have opposite signs, then there exists at least one \(c\) in the interval \((a,b)\) such that \(f(c)=0\) (an \(x -\)intercept).

Step2: Check sign changes

For \(x=-2.3\), \(y = - 0.09\) (negative). For \(x=-1.9\), \(y = 0.44\) (positive). There is a sign - change in the interval \(-2.3

Answer:

\(-2.3 < x < -1.9\)