The domain of the piecewise function is $(-\infty, \infty)$.
a. Graph the function.
b. Use your graph to determine the function's range.
[tex]f(x)=\left\{\begin{array}{lll}
x+2 & \text { if } & x\ \textless \ 1 \\
x-2 & \text { if } & x \geq 1
\end{array}\right.[/tex]
a. Choose the correct graph below.
A.
B.
b. The range of [tex]f(x)[/tex] is $\square$ (Type your answer in interval notation.)
Answer (2)
Analyze the piecewise function: f ( x ) = x + 2 for x < 1 and f ( x ) = x − 2 for x ≥ 1 .
Graph the two parts of the function: a line with an open circle at ( 1 , 3 ) for x < 1 and a line starting at ( 1 , − 1 ) for x ≥ 1 .
Identify the correct graph as A.
Determine the range by combining the ranges of both parts: ( − ∞ , 3 ) ∪ [ − 1 , ∞ ) = ( − ∞ , ∞ ) . The range of f ( x ) is ( − ∞ , ∞ ) .
Explanation
Understanding the Problem We are given a piecewise function and asked to graph it and determine its range. The function is defined as f ( x ) = x + 2 if x < 1 and f ( x ) = x − 2 if xg e q 1 .
Analyzing the Graph - Part 1 First, let's analyze the graph of the function. For x < 1 , f ( x ) = x + 2 . This is a line with slope 1 and y-intercept 2. Since x < 1 , the graph is the part of the line to the left of x = 1 . At x = 1 , f ( 1 ) = 1 + 2 = 3 , but since x < 1 , the point ( 1 , 3 ) is not included in the graph. So there is an open circle at ( 1 , 3 ) .
Analyzing the Graph - Part 2 For xg e q 1 , f ( x ) = x − 2 . This is a line with slope 1 and y-intercept -2. Since xg e q 1 , the graph is the part of the line to the right of x = 1 . At x = 1 , f ( 1 ) = 1 − 2 = − 1 . So the point ( 1 , − 1 ) is included in the graph.
Identifying the Correct Graph Based on this analysis, the correct graph is A.
Determining the Range Now, let's determine the range of the function. For x < 1 , f ( x ) = x + 2 < 1 + 2 = 3 . So the range of this part is ( − ∞ , 3 ) . For xg e q 1 , f ( x ) = x − 2 g e q 1 − 2 = − 1 . So the range of this part is [ − 1 , ∞ ) . The range of the entire function is the union of the ranges of the two parts, which is ( − ∞ , 3 ) ∪ [ − 1 , ∞ ) = ( − ∞ , ∞ ) .
Final Answer Therefore, the range of f ( x ) is ( − ∞ , ∞ ) .
Examples
Piecewise functions are used in real life to model situations where the rule or relationship changes based on the input. For example, cell phone plans often have different rates for data usage depending on whether you are below or above a certain data limit. Similarly, income tax brackets are a piecewise function where the tax rate changes based on your income level. Understanding piecewise functions helps in analyzing and predicting outcomes in these scenarios.
The graph of the piecewise function has an open circle at (1, 3) for the part defined as x < 1 and a solid point at (1, -1) for x ≥ 1. The function's range is (-∞, ∞), which encompasses all real numbers. You will need to identify the correct graph from the options provided based on these characteristics.
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