Which graph is the graph of this function?
[tex]f(x)=\left\{\begin{array}{ll}5 & \text { if }-3 < x < -2 \\-\frac{3}{2} & \text { if }-2 \leq x < 3 \\ \frac{1}{2} & \text { if } 3 \leq x \leq 4\end{array}\right.[/tex]
A. graph A
B. graph B
C. graph C
D. graph D
Answer (2)
The function is a piecewise function defined over three intervals.
For − 3 < x < − 2 , f ( x ) = 5 (horizontal line with open circles at x = − 3 and x = − 2 ).
For − 2 ≤ x < 3 , f ( x ) = − 2 3 (horizontal line with closed circle at x = − 2 and open circle at x = 3 ).
For 3 ≤ x ≤ 4 , f ( x ) = 2 1 (horizontal line with closed circles at x = 3 and x = 4 ).
Graph B matches these characteristics, so the answer is B .
Explanation
Analyze the Piecewise Function We are given a piecewise function and asked to identify its graph. Let's analyze the function interval by interval.
Analyze the First Interval For − 3 < x < − 2 , f ( x ) = 5 . This means we have a horizontal line at y = 5 on this interval. The endpoints x = − 3 and x = − 2 are not included, so we should have open circles at these points.
Analyze the Second Interval For − 2 ≤ x < 3 , f ( x ) = − 2 3 . This is a horizontal line at y = − 2 3 . The endpoint x = − 2 is included (closed circle), and x = 3 is not included (open circle).
Analyze the Third Interval For 3 ≤ x ≤ 4 , f ( x ) = 2 1 . This is a horizontal line at y = 2 1 . Both endpoints x = 3 and x = 4 are included (closed circles).
Compare with the Graphs Now, we need to compare these characteristics to the given graphs (A, B, C, and D) to find the correct one. By carefully examining the graphs based on the intervals and the values of the function, we can determine which graph matches our analysis.
Identify the Correct Graph After comparing the characteristics with the graphs, graph B matches the description of the piecewise function.
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 billing plan might have one rate for the first 100 minutes of calls and a different rate for calls exceeding 100 minutes. 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 modeling such scenarios.
The graph of the piecewise function matches with Graph B, which correctly depicts the defined intervals and values with appropriate open and closed circles. This function has three separate segments with specified conditions for inclusion on the endpoints. Therefore, the answer is B.
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