If [tex]u(x)=-2 x^2+3[/tex] and [tex]v(x)=\frac{1}{x}[/tex], what is the range of [tex](u \circ v)(x)[/tex] ?
A. [tex](\frac{1}{3}, 0)[/tex]
B. [tex](3, \infty)[/tex]
C. [tex](-\infty, 3)[/tex]
D. [tex](-\infty,+\infty)[/tex]
Answer (2)
Find the composite function: ( u ∘ v ) ( x ) = − x 2 2 + 3 .
Analyze the behavior: As x approaches infinity, ( u ∘ v ) ( x ) approaches 3, but never reaches it.
Determine the range: Since x 2 is always positive, ( u ∘ v ) ( x ) is always less than 3.
State the final answer: The range of ( u ∘ v ) ( x ) is ( − ∞ , 3 ) . ( − ∞ , 3 )
Explanation
Find the composite function First, we need to find the composite function ( u ∘ v ) ( x ) . This means we need to substitute v ( x ) into u ( x ) .
Substitute v(x) into u(x) We have u ( x ) = − 2 x 2 + 3 and v ( x ) = x 1 . So, ( u ∘ v ) ( x ) = u ( v ( x )) = u ( x 1 ) = − 2 ( x 1 ) 2 + 3 = − x 2 2 + 3 .
Analyze the behavior of the composite function Now, we need to determine the range of the function ( u ∘ v ) ( x ) = − x 2 2 + 3 . Since x 2 is always positive for any non-zero x , we have 0"> x 2 > 0 for x = 0 . Therefore, 0"> x 2 1 > 0 , and thus, − x 2 2 < 0 .
Determine the upper bound of the range Adding 3 to both sides of the inequality, we get − x 2 2 + 3 < 3 . This means that the value of the composite function is always less than 3.
Analyze the limit as x approaches infinity As x approaches infinity, x 2 also approaches infinity, so x 2 2 approaches 0. Therefore, − x 2 2 + 3 approaches 3. However, since x cannot be zero, the function will never actually reach the value of 3.
Analyze the limit as x approaches 0 As x approaches 0, x 2 approaches 0, so x 2 2 approaches infinity. Therefore, − x 2 2 + 3 approaches negative infinity.
State the range Thus, the range of the function ( u ∘ v ) ( x ) = − x 2 2 + 3 is all real numbers less than 3, which can be written as ( − ∞ , 3 ) .
Examples
Imagine you are designing a lens system where the magnification of one lens affects the input of another. If the first lens has a magnification factor of v ( x ) = x 1 and the second lens adjusts the image according to u ( x ) = − 2 x 2 + 3 , understanding the range of the composite function ( u ∘ v ) ( x ) helps you determine the possible output sizes. This is crucial for ensuring the final image fits within the display or sensor constraints. By analyzing the range, you can predict the minimum and maximum sizes of the final image, which is essential for practical applications in optics and imaging systems.
The range of the composite function ( u ∘ v ) ( x ) is ( − ∞ , 3 ) , as it approaches 3 but never actually reaches that value. Therefore, it includes all real numbers less than 3. The correct choice from the options provided is C: ( − ∞ , 3 ) .
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