**Problem 6: Determining Continuity of $ r(x) $**Determine which values of $ C $ make the function $ r(x) $:- (a) left-continuous,- (b) right-continuous,- (c) continuous.The piecewise function $ r(x) $ is defined as:\[r(x) = \begin{cases} x^2 + 3 & \text{for } x < -1 \\Cx & \text{for } -1 \leq x \leq 2 \\-x^2 + 2 & \text{for } x > 2 \end{cases}\]**Explanation:**- The first piece $ x^2 + 3 $ defines $ r(x) $ for values of $ x $ less than -1.- The second piece $ Cx $ defines $ r(x) $ for $ x $ between -1 and 2, inclusive.- The third piece $-x^2 + 2$ defines $ r(x) $ for values of $ x $ greater than 2.To find the values of $ C $ that make the function $ r(x) $ left-continuous, right-continuous, or continuous, one must ensure the function values and limits match at the boundaries $ x = -1 $ and $ x = 2 $.

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Description: **Problem 6: Determining Continuity of $ r(x) $**Determine which values of $ C $ make the function $ r(x) $:- (a) left-continuous,- (b) right-continuous,- (c) continuous.The piecewise function $ r(x) $ is defined as:\[r(x) = \begin{cases} x^

If the function r(x) is left-continuous, then use limx→-1-rx=limx→-1+rx, to determine the value of…

Answered: Which values of C makes r(x) (a)…

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Which values of C makes r(x) (a) left-continuous, (b) right-continuous (c) continuous x2 +3 for x < -1 r(x) = Cx for –1 2

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Question

**Problem 6: Determining Continuity of $ r(x) $**

Determine which values of $ C $ make the function $ r(x) $:

- (a) left-continuous,
- (b) right-continuous,
- (c) continuous.

The piecewise function $ r(x) $ is defined as:

\[
r(x) =
\begin{cases}
x^2 + 3 & \text{for } x < -1 \\
Cx & \text{for } -1 \leq x \leq 2 \\
-x^2 + 2 & \text{for } x > 2
\end{cases}
\]

**Explanation:**

- The first piece $ x^2 + 3 $ defines $ r(x) $ for values of $ x $ less than -1.
- The second piece $ Cx $ defines $ r(x) $ for $ x $ between -1 and 2, inclusive.
- The third piece $-x^2 + 2$ defines $ r(x) $ for values of $ x $ greater than 2.

To find the values of $ C $ that make the function $ r(x) $ left-continuous, right-continuous, or continuous, one must ensure the function values and limits match at the boundaries $ x = -1 $ and $ x = 2 $.

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