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...
Related questions
Question
![**Graph the Function:**
\[ f(x) =
\begin{cases}
\frac{8}{x-2} & \text{if } x < 2 \\
(x-3)^2 & \text{if } 2 \leq x \leq 4 \\
\frac{8}{4-x} & \text{if } x > 4
\end{cases} \]
**Explanation for Educational Context:**
This piecewise function consists of three different expressions, each applicable to a specific interval of the variable \(x\):
1. **For \(x < 2\):**
- The function is defined by the expression \(\frac{8}{x-2}\). This is a rational function, which may have a vertical asymptote at \(x = 2\) (not defined within the interval given).
2. **For \(2 \leq x \leq 4\):**
- The function is defined by the expression \((x-3)^2\). This is a quadratic function, which represents a parabola opening upwards. The vertex of this parabola is at \(x = 3\).
3. **For \(x > 4\):**
- The function is defined by the expression \(\frac{8}{4-x}\). Similar to the first interval, this is a rational function with a potential vertical asymptote at \(x = 4\) (again, not defined within the interval given).
Teaching students to graph this function involves plotting each piece over its respective interval and understanding how they connect at the boundaries. It is also important to discuss the properties such as continuity and asymptotic behavior.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F42dff072-7d42-411c-b912-034f8f72c397%2F4030eebf-417a-4240-a748-a75039810766%2F3fyov7t_processed.png&w=3840&q=75)
Transcribed Image Text:**Graph the Function:**
\[ f(x) =
\begin{cases}
\frac{8}{x-2} & \text{if } x < 2 \\
(x-3)^2 & \text{if } 2 \leq x \leq 4 \\
\frac{8}{4-x} & \text{if } x > 4
\end{cases} \]
**Explanation for Educational Context:**
This piecewise function consists of three different expressions, each applicable to a specific interval of the variable \(x\):
1. **For \(x < 2\):**
- The function is defined by the expression \(\frac{8}{x-2}\). This is a rational function, which may have a vertical asymptote at \(x = 2\) (not defined within the interval given).
2. **For \(2 \leq x \leq 4\):**
- The function is defined by the expression \((x-3)^2\). This is a quadratic function, which represents a parabola opening upwards. The vertex of this parabola is at \(x = 3\).
3. **For \(x > 4\):**
- The function is defined by the expression \(\frac{8}{4-x}\). Similar to the first interval, this is a rational function with a potential vertical asymptote at \(x = 4\) (again, not defined within the interval given).
Teaching students to graph this function involves plotting each piece over its respective interval and understanding how they connect at the boundaries. It is also important to discuss the properties such as continuity and asymptotic behavior.
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