Find the values of x for which the function is continuous.

Question

Quizplus · Accepted Answer

For a function to be continuous, the limit of the function as x approaches the point of interest must exist and be equal to the value of the function at that point. Looking at the given function, we have a piecewise function: f(x) = 2x + 3, for x < 1 f(x) = 5, for x = 1 f(x) = x^2 - 1, for x > 1 To check for continuity at x = 1, we need to evaluate the limit as x approaches 1 for each piece of the function. lim (x -> 1-) f(x) = lim (x -> 1-) 2x + 3 = 5 lim (x -> 1) f(x) = 5 lim (x -> 1+) f(x) = lim (x -> 1+) x^2 - 1 = 0 Since all three limits exist and are equal to 5, the function is continuous at x = 1. For x < 1 and x > 1, we have polynomial functions which are continuous everywhere. Therefore, the function is continuous for all values of x. The only answer choice that includes all values of x is D.

Find the Values of X for Which the Function Is