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Chapter 1: Problem 27

Let $$ f(x)=\left\\{\begin{array}{ll} x+2 & \text { if } x \leq 1 \\ k x^{2} & \text { if } x>1 \end{array}\right. $$ Find the value of \(k\) that will make \(f\) continuous on \((-\infty, \infty)\).

Short Answer

Expert verified
The value of k that makes f(x) continuous on \((-\infty, \infty)\) is 3.

Step by step solution

01

Find the left-side limit of f(x) as x approaches 1

Since x + 2 is defined for x 鈮� 1, we will find the limit from the left as x approaches 1: \(\lim_{x\to1^-} f(x) = \lim_{x\to1^-} (x+2)\) As x approaches 1 from the left side, we can substitute 1 in the expression: \(\lim_{x\to1^-} f(x) = 1 + 2 = 3\)
02

Find the right-side limit of f(x) as x approaches 1

Since kx^2 is defined for x > 1, we will find the limit from the right as x approaches 1: \(\lim_{x\to1^+} f(x) = \lim_{x\to1^+} (kx^2)\) As x approaches 1 from the right side, we can substitute 1 in the expression: \(\lim_{x\to1^+} f(x) = k(1)^2 = k\)
03

Set the left-side limit equal to the right-side limit to find k

For f(x) to be continuous at x = 1, the left-side limit must equal the right-side limit. Therefore, we set 3 (left-side limit) equal to k (right-side limit): \(3 = k\)
04

Identify the value of k that makes f(x) continuous

The equation 3 = k gives us the value of k that makes f(x) continuous on \((-\infty, \infty)\). Thus: \(k = 3\) The value of k that makes f(x) continuous on \((-\infty, \infty)\) is 3.

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