Explain how to graph the given piecewise-defined function. Be sure to specify the type of endpoint each piece of the function will have and why.
[tex]f(x)=\left\{\begin{array}{ll} -x+3, & x\ \textless \ 2 \\ 3, & 2 \leq x\ \textless \ 4 \\ 4-2 x, & x \geq 4 \end{array}\right.[/tex]
Answer (2)
The function consists of three parts, each defined on a specific interval.
For x < 2 , the function is a line y = − x + 3 with an open circle at (2, 1).
For 2 ≤ x < 4 , the function is a horizontal line y = 3 with a closed circle at (2, 3) and an open circle at (4, 3).
For x ≥ 4 , the function is a line y = 4 − 2 x with a closed circle at (4, -4).
The graph consists of these three pieces plotted on their respective intervals, with attention to open and closed endpoints. The final graph is a combination of these pieces.
Explanation
Understanding the Piecewise Function We are given a piecewise-defined function and asked to explain how to graph it, specifying the type of endpoint each piece will have. The function is defined as: f ( x ) = ⎩ ⎨ ⎧ − x + 3 , 3 , 4 − 2 x , x < 2 2 ≤ x < 4 x ≥ 4
Graphing the First Piece The first piece of the function is f ( x ) = − x + 3 for x < 2 . This is a linear function with a slope of -1 and a y-intercept of 3. To graph this, we can find two points on the line. Since x < 2 , we can choose x = 0 and x = 1 . When x = 0 , f ( 0 ) = − 0 + 3 = 3 . When x = 1 , f ( 1 ) = − 1 + 3 = 2 . So, we have the points (0, 3) and (1, 2). Since the inequality is strict ( x < 2 ), the endpoint at x = 2 will be an open circle. To find the y-value at x = 2 , we plug in x = 2 into the equation: f ( 2 ) = − 2 + 3 = 1 . So, the endpoint will be at (2, 1) and it will be an open circle.
Graphing the Second Piece The second piece of the function is f ( x ) = 3 for 2 ≤ x < 4 . This is a horizontal line at y = 3 . Since the inequality includes x = 2 ( 2 ≤ x ), the endpoint at x = 2 will be a closed circle. The y-value is 3, so the endpoint is (2, 3). Since the inequality is strict at x = 4 ( x < 4 ), the endpoint at x = 4 will be an open circle. The y-value is 3, so the endpoint is (4, 3).
Graphing the Third Piece The third piece of the function is f ( x ) = 4 − 2 x for x ≥ 4 . This is a linear function with a slope of -2 and a y-intercept of 4. Since the inequality includes x = 4 ( x ≥ 4 ), the endpoint at x = 4 will be a closed circle. To find the y-value at x = 4 , we plug in x = 4 into the equation: f ( 4 ) = 4 − 2 ( 4 ) = 4 − 8 = − 4 . So, the endpoint will be at (4, -4) and it will be a closed circle. To graph this, we can find another point on the line. Let's choose x = 5 . Then f ( 5 ) = 4 − 2 ( 5 ) = 4 − 10 = − 6 . So, we have the point (5, -6).
Summary of the Graph In summary, we have three pieces:
A line y = − x + 3 for x < 2 , with an open circle at (2, 1).
A horizontal line y = 3 for 2 ≤ x < 4 , with a closed circle at (2, 3) and an open circle at (4, 3).
A line y = 4 − 2 x for x ≥ 4 , with a closed circle at (4, -4).
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 a classic example of a piecewise function, where different tax rates apply to different income ranges. Understanding how to graph and analyze piecewise functions helps in understanding these real-world scenarios and making informed decisions.
To graph the piecewise function, plot three separate pieces with specified endpoints. The first piece has an open circle at ( 2 , 1 ) , the second has a closed circle at ( 2 , 3 ) and an open circle at ( 4 , 3 ) , and the third has a closed circle at ( 4 , − 4 ) . Finally, connect the pieces according to the intervals they cover.
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