Which graph represents the piecewise-defined function [tex]f(x)=\left\{\begin{array}{ll}-x+4, & 0 \leq x < 3 \\ 6, & x \geq 3\end{array} ?\right.[/tex]
Answer (2)
The function is a piecewise-defined function, with f ( x ) = − x + 4 for 0 ≤ x < 3 and f ( x ) = 6 for x ≥ 3 .
The first piece is a line segment starting at ( 0 , 4 ) and approaching ( 3 , 1 ) with an open circle.
The second piece is a horizontal line at y = 6 for x ≥ 3 , starting with a closed circle at ( 3 , 6 ) .
By analyzing these characteristics, we can identify the correct graph.
Explanation
Analyzing the Piecewise Function We are given a piecewise-defined function and asked to identify its graph. The function is defined as follows:
f ( x ) = { − x + 4 , 6 , 0 ≤ x < 3 x ≥ 3
We need to analyze the behavior of each piece of the function to determine the correct graph.
Analyzing the First Piece The first piece of the function is f ( x ) = − x + 4 for 0 ≤ x < 3 . This is a linear function with a slope of -1 and a y-intercept of 4. Let's find the value of the function at the endpoints of the interval.
At x = 0 , f ( 0 ) = − 0 + 4 = 4 . So, the graph starts at the point ( 0 , 4 ) .
As x approaches 3 from the left, f ( x ) approaches − 3 + 4 = 1 . Since the interval is x < 3 , the point ( 3 , 1 ) is not included in the graph of this piece. This means there should be an open circle at ( 3 , 1 ) .
Analyzing the Second Piece The second piece of the function is f ( x ) = 6 for x ≥ 3 . This is a horizontal line at y = 6 . Let's find the value of the function at x = 3 .
At x = 3 , f ( 3 ) = 6 . So, the graph includes the point ( 3 , 6 ) . This means there should be a closed circle at ( 3 , 6 ) .
For all 3"> x > 3 , the function remains at f ( x ) = 6 , so the graph is a horizontal line at y = 6 for x ≥ 3 .
Conclusion Based on our analysis:
The graph starts at ( 0 , 4 ) and has a slope of -1 until x = 3 , approaching the point ( 3 , 1 ) with an open circle.
At x = 3 , the graph jumps to y = 6 and continues as a horizontal line for x ≥ 3 .
Examples
Piecewise functions are used in real life to model situations where different rules or conditions apply over different intervals. For example, a cell phone plan might charge one rate for the first 100 minutes and a different rate for each additional minute. Similarly, income tax brackets are defined using a piecewise function, where the tax rate changes as income increases. Understanding piecewise functions helps in analyzing and predicting outcomes in these scenarios.
The piecewise function shows a linear segment from (0, 4) to an open circle at (3, 1), then a constant value at y = 6 starting with a closed circle at (3, 6). This results in a graph that drops to y = 1 before x = 3 and jumps to y = 6 at x = 3 with a horizontal line thereafter.
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