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 (2)
The piecewise function consists of three parts: a line for x < − 2 with an open circle at ( − 2 , 2 ) , a constant line at y = 2 from − 2 to 3 with closed circles at both endpoints, and a line for 3"> x > 3 approaching an open circle at ( 3 , 3 ) . Thus, you should look for a graph that reflects these features accurately. The correct graph is the one that matches these descriptions.
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The function consists of three parts: a line with slope 2 1 for x < − 2 , a horizontal line at y = 2 for − 2 ≤ x ≤ 3 , and a line with slope 2 for 3"> x > 3 .
At x = − 2 , the first part approaches y = 2 with an open circle, while the second part has a closed circle at y = 2 .
At x = 3 , the second part has a closed circle at y = 2 , while the third part approaches y = 3 with an open circle.
The graph that represents this piecewise function will have these characteristics. The correct graph is the one that matches these features.
Explanation
Understanding the Piecewise Function We are given a piecewise function and asked to identify its graph. The function is defined as follows:
3 \end{array}\right."> g ( x ) = ⎩ ⎨ ⎧ 2 1 x + 3 , 2 , 2 x − 3 , x < − 2 − 2 ≤ x ≤ 3 x > 3
We need to analyze each piece of the function and how they connect at the boundaries x = − 2 and x = 3 .
Analyzing the First Piece Let's analyze the first piece, g ( x ) = 2 1 x + 3 for x < − 2 . This is a linear function with a slope of 2 1 and a y-intercept of 3. As x approaches − 2 from the left, g ( x ) approaches 2 1 ( − 2 ) + 3 = − 1 + 3 = 2 . Since the inequality is strict ( x < − 2 ), there should be an open circle at the point ( − 2 , 2 ) .
Analyzing the Second Piece Now consider the second piece, g ( x ) = 2 for − 2 ≤ x ≤ 3 . This is a constant function, meaning g ( x ) is always 2 within this interval. At x = − 2 , g ( − 2 ) = 2 , so there is a closed circle at ( − 2 , 2 ) . At x = 3 , g ( 3 ) = 2 , so there is a closed circle at ( 3 , 2 ) .
Analyzing the Third Piece Finally, let's analyze the third piece, g ( x ) = 2 x − 3 for 3"> x > 3 . This is a linear function with a slope of 2 and a y-intercept of -3. As x approaches 3 from the right, g ( x ) approaches 2 ( 3 ) − 3 = 6 − 3 = 3 . Since the inequality is strict ( 3"> x > 3 ), there should be an open circle at the point ( 3 , 3 ) .
Summary of Graph Features In summary, the graph should have:
A line with slope 2 1 for x < − 2 , approaching ( − 2 , 2 ) with an open circle.
A horizontal line at y = 2 for − 2 ≤ x ≤ 3 , with closed circles at ( − 2 , 2 ) and ( 3 , 2 ) .
A line with slope 2 for 3"> x > 3 , approaching ( 3 , 3 ) with an open circle.
Examples
Piecewise functions are used in real life to model situations where the rule or relationship changes based on the input. For example, a cell phone billing plan might charge one rate for the first 100 minutes and a different rate for each minute thereafter. 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.
