given the values in the table on the left come from a continuous function, which intervals must contain an x…

given the values in the table on the left come from a continuous function, which intervals must contain an x - intercept? select two options.\n-0.8 < x < 0\n-3.2 < x < -2.4\n2.4 < x < 3.2\n1.6 < x < 2.4\ndone

given the values in the table on the left come from a continuous function, which intervals must contain an x - intercept? select two options.\n-0.8 < x < 0\n-3.2 < x < -2.4\n2.4 < x < 3.2\n1.6 < x < 2.4\ndone

Answer

Explanation:

Step1: Use 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 an (x -)intercept, (k = 0). So we need to find intervals where (y) changes sign (from positive to negative or vice - versa).

Step2: Check each interval

  • For the interval (-3.2\lt x\lt - 2.4): When (x=-3.2), (y = f(-3.2)=-1.19) (negative), and when (x=-2.4), (y = f(-2.4)=4.62) (positive). Since the function is continuous, by the Intermediate Value Theorem, there is an (x) - intercept in the interval ((-3.2,-2.4)) because (y) changes sign from negative to positive.

  • For the interval (1.6\lt x\lt2.4): When (x = 1.6), (y=f(1.6)=-4.7) (negative), and when (x = 2.4), (y=f(2.4)=7.26) (positive). Since the function is continuous, by the Intermediate Value Theorem, there is an (x) - intercept in the interval ((1.6,2.4)) because (y) changes sign from negative to positive.

  • For the interval (-0.8\lt x\lt0): When (x=-0.8), (y = f(-0.8)=-0.64) (negative), and when (x = 0), (y=f(0)=-5.58) (negative). There is no sign change.

  • For the interval (2.4\lt x\lt3.2): When (x = 2.4), (y=f(2.4)=7.26) (positive), and when (x = 3.2), (y=f(3.2)=30.99) (positive). There is no sign change.

Answer:

(-3.2\lt x\lt - 2.4), (1.6\lt x\lt2.4)