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 g ( x ) is a piecewise function defined differently for x < 0 , 04 .
For x < 0 , g ( x ) = x 2 represents the left half of a parabola.
For 04 , g ( x ) = x is a line starting from ( 4 , 4 ) (exclusive) with a slope of 1.
The graph that matches these conditions represents the piecewise function.
Explanation
Analyze the piecewise function We are given a piecewise function and asked to identify its graph. The function is defined as follows:
g ( x ) = { x 2 , 2 1 x , x < 0 04
We need to analyze each piece of the function and see how it looks on the graph.
Analyze the first piece For x < 0 , g ( x ) = x 2 . This is a parabola opening upwards, but only defined for negative values of x . So, we should see the left half of a parabola in the second quadrant.
Analyze the second piece For 0 < x ≤ 4 , g ( x ) = 2 1 x . This is a linear function with a slope of 2 1 . It starts at x = 0 (but does not include it, since the inequality is strict 0 < x ) and ends at x = 4 (including it, since x ≤ 4 ). When x = 4 , g ( 4 ) = 2 1 × 4 = 2 . So, we have a line segment from ( 0 , 0 ) (not included) to ( 4 , 2 ) (included).
Analyze the third piece For 4"> x > 4 , g ( x ) = x . This is a linear function with a slope of 1 . It starts at x = 4 (but does not include it, since the inequality is strict 4"> x > 4 ). When x = 4 , g ( 4 ) = 4 . So, we have a line starting from ( 4 , 4 ) (not included) and going to infinity with a slope of 1.
Find the matching graph Now, we need to find the graph that matches these three conditions:
Left half of a parabola in the second quadrant.
A line segment from ( 0 , 0 ) (not included) to ( 4 , 2 ) (included).
A line starting from ( 4 , 4 ) (not included) and going to infinity with a slope of 1.
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 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 has three parts: a parabola for x < 0 , a line from 0 < x ≤ 4 , and another line for 4"> x > 4 . The graph includes the left half of a parabola, a line segment from just above (0,0) to (4,2), and a line starting just above (4,4). By understanding these pieces, we can accurately identify the corresponding graph for the function.
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