Question

Find the constant $a$, or the constants $a$ and $b$, such that the function is continuous on the entire real number line. $$f(x)=\left\{\begin{array}{ll} 2, & x \leq-1 \\ a x+b, & -1<x<3 \\ -2, & x \geq 3 \end{array}\right.$$

Name: Problem 65, Chapter 2: Calculus Early Transcendental Functions
Uploaded: 2021-07-10T23:02:20-08:00
Duration: 2 min 32 s
Channel: Noel Aguilar
Description: Find the constant a, or the constants a and b, such that the function is continuous on the entire real number line. f(x)={ 2, x ≤-1 a x+b, -1<x<3 -2, x ≥ 3 .

   Find the constant $a$, or the constants $a$ and $b$, such that the function is continuous on the entire real number line.
$$f(x)=\left\{\begin{array}{ll}
2, & x \leq-1 \\
a x+b, & -1<x<3 \\
-2, & x \geq 3
\end{array}\right.$$

Calculus Early Transcendental Functions

Ron Larson, Bruce… 6th Edition

Chapter 2, Problem 65

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Solved by Expert Noel Aguilar

07/10/2021

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Find the constant $a$, or the constants $a$ and $b$, such that the function is continuous on the entire real number line. $$f(x)=\left\{\begin{array}{ll} 2, & x \leq-1 \\ a x+b, & -1<x<3 \\ -2, & x \geq 3 \end{array}\right.$$

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Key Concepts

Piecewise Functions

Piecewise functions are defined by different expressions depending on the interval of the domain they are evaluated in. When working with piecewise functions, it is critical to consider the behavior of the function in each separate region and also examine the points where the pieces meet to ensure that the function behaves as intended across the entire domain.

Continuity at Transition Points

Continuity at a transition point means that the limit of the function from the left side of the point equals the limit from the right side, and both equal the function’s actual value at that point if it is defined there. For piecewise functions, this typically involves setting the function values from adjacent pieces equal at their shared endpoints, thereby ensuring there is no jump or break in the graph.

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