Graph the following function. State the domain and range in interval notation. Determine the x<1 intervals where the function is increasing, decreasing, or constant. g(x) = x2 1sxs2 x>2
Graph the following function. State the domain and range in interval notation. Determine the x<1 intervals where the function is increasing, decreasing, or constant. g(x) = x2 1sxs2 x>2
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Exponents
The exponent or power or index of a variable/number is the number of times that variable/number is multiplied by itself.
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![**Graphing and Analyzing Piecewise Functions**
In this example, we are given a piecewise function and asked to graph it, state its domain and range, and understand its behavior in terms of whether it is increasing, decreasing, or constant.
**Piecewise Function:**
\[ g(x) = \begin{cases}
1 & \text{if } x < 1 \\
x^2 & \text{if } 1 \leq x \leq 2 \\
4 & \text{if } x > 2
\end{cases} \]
### Steps to Graph the Function
1. **For \( x < 1 \):**
- The function \( g(x) = 1 \) is a constant value of 1.
- Plot a horizontal line at \( y = 1 \) for \( x \)-values less than 1.
2. **For \( 1 \leq x \leq 2 \):**
- The function \( g(x) = x^2 \) follows a parabolic curve.
- Plot the points from the equation \( y = x^2 \) within the interval [1, 2].
3. **For \( x > 2 \):**
- The function \( g(x) = 4 \) is another constant value.
- Plot a horizontal line at \( y = 4 \) for \( x \)-values greater than 2.
### Domain and Range
- **Domain:** The domain of \( g(x) \) is all real numbers since the function is defined for every \( x \)-value.
In interval notation: \(( -\infty, \infty )\)
- **Range:** The function takes on three different ranges associated with each piece of the function.
In interval notation: \( \{1\} \cup [1, 4] \cup \{4\} = [1, 4] \)
### Behavior of the Function
- **Increasing, Decreasing, or Constant Intervals:**
- **Constant:** For \( x < 1 \), \( g(x) = 1 \).
- **Increasing:** For \( 1 \leq x \leq 2 \), \( g(x) = x^2 \) is increasing.
- **Constant:** For \( x >](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7921e840-b4be-4997-af5c-a45fa0acee7f%2Fda8e3533-5848-4a1d-96c7-aa2396cb5c44%2Feob7n54_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Graphing and Analyzing Piecewise Functions**
In this example, we are given a piecewise function and asked to graph it, state its domain and range, and understand its behavior in terms of whether it is increasing, decreasing, or constant.
**Piecewise Function:**
\[ g(x) = \begin{cases}
1 & \text{if } x < 1 \\
x^2 & \text{if } 1 \leq x \leq 2 \\
4 & \text{if } x > 2
\end{cases} \]
### Steps to Graph the Function
1. **For \( x < 1 \):**
- The function \( g(x) = 1 \) is a constant value of 1.
- Plot a horizontal line at \( y = 1 \) for \( x \)-values less than 1.
2. **For \( 1 \leq x \leq 2 \):**
- The function \( g(x) = x^2 \) follows a parabolic curve.
- Plot the points from the equation \( y = x^2 \) within the interval [1, 2].
3. **For \( x > 2 \):**
- The function \( g(x) = 4 \) is another constant value.
- Plot a horizontal line at \( y = 4 \) for \( x \)-values greater than 2.
### Domain and Range
- **Domain:** The domain of \( g(x) \) is all real numbers since the function is defined for every \( x \)-value.
In interval notation: \(( -\infty, \infty )\)
- **Range:** The function takes on three different ranges associated with each piece of the function.
In interval notation: \( \{1\} \cup [1, 4] \cup \{4\} = [1, 4] \)
### Behavior of the Function
- **Increasing, Decreasing, or Constant Intervals:**
- **Constant:** For \( x < 1 \), \( g(x) = 1 \).
- **Increasing:** For \( 1 \leq x \leq 2 \), \( g(x) = x^2 \) is increasing.
- **Constant:** For \( x >
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