Which graph represents the piecewise-defined function [tex]f(x)=\left\{\begin{array}{ll}-1.5 x+3.5, & x\ \textless \ 2 \\ 4+x, & x \geq 2\end{array}\right.[/tex]?
Answer (2)
The function is a piecewise-defined function with two linear pieces.
For x < 2 , f ( x ) = − 1.5 x + 3.5 , which is a line with a slope of -1.5 and passes through (0, 3.5) and approaches (2, 0.5) with an open circle.
For x ≥ 2 , f ( x ) = 4 + x , which is a line with a slope of 1 and passes through (2, 6) with a closed circle.
The graph that represents this piecewise function has the described characteristics.
Explanation
Analyzing the Piecewise Function We are given a piecewise-defined function: f ( x ) = { − 1.5 x + 3.5 , 4 + x , x < 2 x ≥ 2 We need to identify the graph that represents this function. Let's analyze the function piece by piece.
Analyzing the First Piece For x < 2 , the function is f ( x ) = − 1.5 x + 3.5 . This is a linear function with a slope of − 1.5 and a y-intercept of 3.5 . When x = 0 , f ( 0 ) = − 1.5 ( 0 ) + 3.5 = 3.5 . When x = 2 , f ( 2 ) = − 1.5 ( 2 ) + 3.5 = − 3 + 3.5 = 0.5 . Since this piece is defined for x < 2 , it approaches the point ( 2 , 0.5 ) but does not include it. Therefore, the graph should have an open circle at ( 2 , 0.5 ) .
Analyzing the Second Piece For x ≥ 2 , the function is f ( x ) = 4 + x . This is a linear function with a slope of 1 and a y-intercept of 4 . When x = 2 , f ( 2 ) = 4 + 2 = 6 . Since this piece is defined for x ≥ 2 , it includes the point ( 2 , 6 ) . Therefore, the graph should have a closed circle at ( 2 , 6 ) .
Finding the Matching Graph Based on our analysis:
The first piece is a line with a slope of − 1.5 and a y-intercept of 3.5 , defined for x < 2 , with an open circle at ( 2 , 0.5 ) .
The second piece is a line with a slope of 1 , defined for x ≥ 2 , with a closed circle at ( 2 , 6 ) .
Now, we need to find the graph that matches these characteristics.
Examples
Piecewise functions are used in real life to model situations where the rule or relationship changes based on the input value. 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 as your income increases. Understanding piecewise functions helps in analyzing and predicting outcomes in these scenarios.
The piecewise function has two parts: a decreasing line for x < 2 with an open circle at ( 2 , 0.5 ) , and an increasing line for x ≥ 2 with a closed circle at ( 2 , 6 ) . Identify the graph that accurately reflects these features. Look for the correct open and closed circles at the specified coordinates.
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