If $u(x)=-2 x^2+3$ and $v(x)=\frac{1}{x}$, what is the range of $(u \circ v)(x)$?
A. $\left(\frac{1}{3}, 0\right)$
B. $(3, \infty)$
C. $(-\infty, 3)$
D. $(-\infty,+\infty)$
Answer (2)
Find the composite function ( u c i rc v ) ( x ) = − f r a c 2 x 2 + 3 .
Determine the domain of the composite function: all real numbers except x = 0 .
Analyze the behavior of the function as x approaches 0 and p m in f t y .
Determine the range of the function: ( − in f t y , 3 ) .
Explanation
Understanding the Problem We are given two functions, u ( x ) = − 2 x 2 + 3 and v ( x ) = f r a c 1 x , and we want to find the range of the composite function ( u c i rc v ) ( x ) . This means we need to find all possible output values of the function when we plug in all possible input values.
Finding the Composite Function First, we need to find the composite function ( u c i rc v ) ( x ) . This means we need to substitute v ( x ) into u ( x ) . So, we have: ( u c i rc v ) ( x ) = u ( v ( x )) = u ( f r a c 1 x ) = − 2 ( f r a c 1 x ) 2 + 3 = − f r a c 2 x 2 + 3.
Determining the Domain Now we need to determine the domain of the composite function. Since v ( x ) = f r a c 1 x , x cannot be 0. Therefore, the domain of ( u c i rc v ) ( x ) is all real numbers except x = 0 .
Analyzing the Function's Behavior Next, we analyze the behavior of the function f ( x ) = − f r a c 2 x 2 + 3 . As x approaches 0, x 2 approaches 0, so f r a c 1 x 2 approaches in f t y . Thus, − f r a c 2 x 2 approaches − in f t y . Therefore, as x approaches 0, f ( x ) approaches − in f t y .
As x approaches p m in f t y , f r a c 1 x 2 approaches 0, so − f r a c 2 x 2 approaches 0. Thus, f ( x ) approaches 3.
Determining the Range Since 0"> x 2 > 0 for all x n e q 0 , we have 0"> f r a c 2 x 2 > 0 , so − f r a c 2 x 2 < 0 . Therefore, − f r a c 2 x 2 + 3 < 3 . This means that the function f ( x ) = − f r a c 2 x 2 + 3 can take any value less than 3, but it can never reach 3.
Stating the Range Therefore, the range of f ( x ) is ( − in f t y , 3 ) .
Examples
Understanding function composition and ranges is crucial in fields like physics and engineering. For instance, consider a scenario where the intensity of light ( u ( x ) ) depends on the distance ( x ) from a light source, and the distance itself is a function of time ( v ( x ) ). Determining the composite function ( u c i rc v ) ( x ) allows us to analyze how the light intensity changes over time. Finding the range of this composite function helps us understand the possible light intensity values that can be observed, which is essential for designing lighting systems or analyzing sensor data.
The composite function is ( u ∘ v ) ( x ) = − x 2 2 + 3 . Its range is determined to be all values less than 3, specifically ( − ∞ , 3 ) . Thus, the correct answer is C. ( − ∞ , 3 ) .
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