The domain of the piecewise function is $(-\infty, \infty)$.
a. Graph the function.
b. Use your graph to determine the function's range.
[tex]f(x)=\left\{\begin{array}{ccc}
\frac{1}{2} x^2 & \text { if } & x\ \textless \ 2 \\
4 x-5 & \text { if } & x \geq 2
\end{array}\right.[/tex]
Answer (2)
Analyze the first piece of the function, f ( x ) = 2 1 x 2 for x < 2 , which is a parabola with a range of [ 0 , 2 ) .
Analyze the second piece of the function, f ( x ) = 4 x − 5 for x ≥ 2 , which is a linear function with a range of [ 3 , ∞ ) .
Combine the ranges of the two pieces to find the overall range.
The range of the piecewise function is [ 0 , 2 ) ∪ [ 3 , ∞ ) . [ 0 , 2 ) ∪ [ 3 , ∞ )
Explanation
Understanding the Problem We are given a piecewise function and asked to graph it and determine its range. The function is defined as: f ( x ) = { 2 1 x 2 4 x − 5 if if x < 2 x ≥ 2 We need to analyze each piece of the function to graph it and then find the range.
Analyzing the First Piece First, let's analyze the first piece, f ( x ) = 2 1 x 2 for x < 2 . This is a parabola with its vertex at ( 0 , 0 ) . Since x < 2 , we only consider the part of the parabola to the left of x = 2 . As x approaches 2, f ( x ) approaches 2 1 ( 2 ) 2 = 2 . So, the function values for this piece range from 0 (inclusive) to 2 (exclusive), i.e., [ 0 , 2 ) .
Analyzing the Second Piece Next, let's analyze the second piece, f ( x ) = 4 x − 5 for x ≥ 2 . This is a linear function with a slope of 4 and a y-intercept of -5. When x = 2 , f ( 2 ) = 4 ( 2 ) − 5 = 3 . Since the slope is positive, as x increases, f ( x ) also increases. Thus, the function values for this piece range from 3 (inclusive) to infinity, i.e., [ 3 , ∞ ) .
Combining the Ranges Now, let's combine the ranges of the two pieces. The first piece has a range of [ 0 , 2 ) , and the second piece has a range of [ 3 , ∞ ) . The overall range of the piecewise function is the union of these two ranges: [ 0 , 2 ) ∪ [ 3 , ∞ ) .
Final Answer Therefore, the range of the given piecewise function is [ 0 , 2 ) ∪ [ 3 , ∞ ) .
Examples
Piecewise functions are used in real life to model situations where different rules or conditions apply over different intervals. For example, consider a cell phone billing plan where the cost per minute is different depending on whether the call is made during peak hours or off-peak hours. Another example is income tax brackets, where different tax rates apply to different income levels. Understanding piecewise functions helps in analyzing and modeling such scenarios accurately.
The piecewise function has two segments: a quadratic function for x < 2 with a range of [ 0 , 2 ) and a linear function for x ≥ 2 with a range of [ 3 , ∞ ) . Thus, the overall range of the function is [ 0 , 2 ) ∪ [ 3 , ∞ ) . The graph reinforces these findings by displaying the characteristics of each segment accurately.
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