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)
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Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
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Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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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 >
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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