The function $h(x)$ is defined as shown.
$h(x)=\left\{\begin{array}{ll}
x+2, & x<3 \
-x+8, & x \geq 3
\end{array}\right.$
What is the range of $h(x)$ ?
A. $-\infty < h(x) < \infty$
B. $h(x) \leq 5$
C. $h(x) \geq 5$
D. $h(x) \geq 3$
Answer (2)
Analyze the first piece of the function, h ( x ) = x + 2 for x < 3 , and determine that h ( x ) < 5 .
Analyze the second piece of the function, h ( x ) = − x + 8 for x ≥ 3 , and determine that h ( x ) ≤ 5 .
Combine the results from both pieces to find the overall range of h ( x ) .
The range of h ( x ) is h ( x ) ≤ 5 , so the final answer is h ( x ) ≤ 5 .
Explanation
Understanding the Problem We are given a piecewise function h ( x ) and we want to find its range. The function is defined as:
h ( x ) = { x + 2 , − x + 8 , x < 3 x ≥ 3
We need to analyze each piece of the function to determine the overall range.
Analyzing the first piece First, let's consider the piece h ( x ) = x + 2 for x < 3 . As x approaches 3 from the left, h ( x ) approaches 3 + 2 = 5 . Since x is strictly less than 3, h ( x ) will always be strictly less than 5. Thus, for this piece, h ( x ) < 5 .
Analyzing the second piece Now, let's consider the piece h ( x ) = − x + 8 for x ≥ 3 . When x = 3 , h ( 3 ) = − 3 + 8 = 5 . As x increases beyond 3, − x + 8 will decrease. For example, when x = 4 , h ( 4 ) = − 4 + 8 = 4 . When x = 10 , h ( 10 ) = − 10 + 8 = − 2 . As x approaches infinity, h ( x ) approaches negative infinity. Thus, for this piece, h ( x ) ≤ 5 .
Combining the results Combining the two pieces, we see that the first piece gives us h ( x ) < 5 , and the second piece gives us h ( x ) ≤ 5 . Since the second piece includes the value 5, the overall range of h ( x ) is all values less than or equal to 5.
Final Answer Therefore, the range of h ( x ) is h ( x ) ≤ 5 .
Examples
Understanding the range of a piecewise function is useful in various real-world scenarios. For example, consider a cell phone plan where the cost is calculated differently based on the data usage. If the usage is below a certain threshold, the cost is calculated one way, and if it exceeds that threshold, the cost is calculated differently. Determining the range of possible costs helps consumers understand the potential expenses they might incur. Similarly, in physics, piecewise functions can model situations where the force acting on an object changes depending on its position or velocity. Analyzing the range of these forces can help predict the object's behavior.
The range of the piecewise function h ( x ) is determined by analyzing both pieces of the function. For x < 3 , h ( x ) < 5 , and for x ≥ 3 , h ( x ) ≤ 5 . Therefore, the overall range of h ( x ) is h ( x ) ≤ 5 .
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