Which graph represents the following piecewise defined function?
[tex]g(x)=\left\{\begin{array}{cc}x^2, & x \ \textless \ 0 \\
\frac{1}{2} x, & 0 \ \textless \ x \leq 4 \\
x, & x \ \textgreater \ 4\end{array}\right.[/tex]
Answer (2)
The function is a parabola x 2 for x < 0 .
It is a line 2 1 x for 04 .
There is a discontinuity at x = 4 . The graph should reflect these features.
Explanation
Analyze the Piecewise Function We are given a piecewise function and asked to identify its graph. Let's analyze each piece of the function to understand its behavior.
Analyze the First Piece For x < 0 , the function is defined as g ( x ) = x 2 . This is a parabola opening upwards, and since x < 0 , we only consider the part of the parabola to the left of the y-axis.
Analyze the Second Piece For 0 < x ≤ 4 , the function is defined as g ( x ) = 2 1 x . This is a linear function with a slope of 2 1 . When x approaches 0 from the right, g ( x ) approaches 0. When x = 4 , g ( 4 ) = 2 1 ( 4 ) = 2 . So, this piece is a line segment starting close to 0 and ending at the point ( 4 , 2 ) .
Analyze the Third Piece For 4"> x > 4 , the function is defined as g ( x ) = x . This is a linear function with a slope of 1. When x approaches 4 from the right, g ( x ) approaches 4. So, this piece is a line starting at the point ( 4 , 4 ) and continuing with a slope of 1.
Summary of the Function's Behavior Now, let's summarize the key features:
For x < 0 , g ( x ) = x 2 (parabola opening upwards, left of y-axis).
For 0 < x ≤ 4 , g ( x ) = 2 1 x (line segment from approximately ( 0 , 0 ) to ( 4 , 2 ) ).
For 4"> x > 4 , g ( x ) = x (line starting at ( 4 , 4 ) with a slope of 1).
Examples
Piecewise functions are used in real life to model situations where different rules 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 minute thereafter. Similarly, income tax brackets are 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 g ( x ) exhibits a parabolic segment for x < 0 , a linear segment from ( 0 , 0 ) to ( 4 , 2 ) for 0 < x ≤ 4 , and a linear segment starting at ( 4 , 4 ) for 4"> x > 4 . Note that there is a discontinuity at x = 4. Therefore, choose a graph that reflects these three behaviors properly.
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