Which graph represents the following piecewise defined function?
[tex]g(x)=\left\{\begin{array}{cl}\frac{1}{2} x+3, & x\ \textless \ -2 \\ 2, & -2 \leq x \leq 3 \\ 2 x-3, & x\ \textgreater \ 3\end{array}\right.[/tex]
Answer (1)
The function is a piecewise function with three parts.
For x < − 2 , it's a line with slope 2 1 and an open circle at ( − 2 , 2 ) .
For − 2 ≤ x ≤ 3 , it's a horizontal line y = 2 with closed circles at ( − 2 , 2 ) and ( 3 , 2 ) .
For 3"> x > 3 , it's a line with slope 2 and an open circle at ( 3 , 3 ) .
The graph representing this piecewise function has these characteristics.
Explanation
Analyze the piecewise function We want to determine the graph that represents the piecewise function: 3 \end{array}\right."> g ( x ) = ⎩ ⎨ ⎧ 2 1 x + 3 , 2 , 2 x − 3 , x < − 2 − 2 ≤ x ≤ 3 x > 3 We will analyze each piece of the function to determine the key features of the graph.
Analyze the first piece For x < − 2 , g ( x ) = 2 1 x + 3 . This is a linear function with a slope of 2 1 and a y-intercept of 3. At x = − 2 , the value of the function is g ( − 2 ) = 2 1 ( − 2 ) + 3 = − 1 + 3 = 2 . Since this interval is defined for x < − 2 , the point ( − 2 , 2 ) will be an open circle.
Analyze the second piece For − 2 ≤ x ≤ 3 , g ( x ) = 2 . This is a constant function, which means the graph will be a horizontal line at y = 2 . At x = − 2 , g ( − 2 ) = 2 , and at x = 3 , g ( 3 ) = 2 . Since the interval is − 2 ≤ x ≤ 3 , the points ( − 2 , 2 ) and ( 3 , 2 ) will be closed circles.
Analyze the third piece For 3"> x > 3 , g ( x ) = 2 x − 3 . This is a linear function with a slope of 2 and a y-intercept of -3. At x = 3 , the value of the function is g ( 3 ) = 2 ( 3 ) − 3 = 6 − 3 = 3 . Since this interval is defined for 3"> x > 3 , the point ( 3 , 3 ) will be an open circle.
Conclusion Based on the analysis:
For x < − 2 , the graph is a line with slope 2 1 passing through ( − 2 , 2 ) with an open circle.
For − 2 ≤ x ≤ 3 , the graph is a horizontal line at y = 2 with closed circles at ( − 2 , 2 ) and ( 3 , 2 ) .
For 3"> x > 3 , the graph is a line with slope 2 passing through ( 3 , 3 ) with an open circle.
Therefore, the graph that represents this piecewise function will have these characteristics.
Examples
Piecewise functions are used in real life to model situations where different rules or conditions apply over different intervals. For example, cell phone plans often have different rates for data usage depending on the amount of data used. The cost might be a fixed amount for the first few gigabytes, then increase at a different rate for additional usage. Similarly, income tax brackets are defined using a piecewise function, where different tax rates apply to different income ranges. Understanding piecewise functions helps in analyzing and predicting outcomes in these scenarios.
