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If [tex]$u(x) = -2x^2 + 3$[/tex] and [tex]$v(x) = \frac{1}{x}$[/tex], what is the range of [tex][tex]$(u \circ v)(x)$[/tex][/tex]?

A. [tex]\left(\frac{1}{3}, 0\right)[/tex]

B. [tex](3, \infty)[/tex]

C. [tex](-\infty, 3)[/tex]

D. [tex](-\infty, +\infty)[/tex]


Sagot :

To find the range of the composite function [tex]\((u \circ v)(x)\)[/tex], we need to determine the behavior of [tex]\((u \circ v)(x)\)[/tex] for all values of [tex]\(x\)[/tex] within the domain of [tex]\(v(x)\)[/tex]. Here, we have [tex]\(u(x) = -2x^2 + 3\)[/tex] and [tex]\(v(x) = \frac{1}{x}\)[/tex].

Step-by-step solution:

1. Find the composite function [tex]\( (u \circ v)(x) \)[/tex]:
[tex]\[ (u \circ v)(x) = u(v(x)) = u\left(\frac{1}{x}\right) \][/tex]

2. Substitute [tex]\( v(x) = \frac{1}{x} \)[/tex] into [tex]\( u(x) \)[/tex]:
[tex]\[ u\left(\frac{1}{x}\right) = -2\left(\frac{1}{x}\right)^2 + 3 \][/tex]

3. Simplify the expression:
[tex]\[ u\left(\frac{1}{x}\right) = -2\left(\frac{1}{x^2}\right) + 3 = -\frac{2}{x^2} + 3 \][/tex]

4. Determine the behavior of [tex]\( -\frac{2}{x^2} + 3 \)[/tex]:
- As [tex]\( x \)[/tex] approaches [tex]\(\infty\)[/tex] or [tex]\(-\infty\)[/tex], [tex]\(\frac{1}{x}\)[/tex] approaches [tex]\(0\)[/tex], so [tex]\(\frac{1}{x^2}\)[/tex] also approaches [tex]\(0\)[/tex], making [tex]\(-\frac{2}{x^2} + 3\)[/tex] approach [tex]\(3\)[/tex].
- As [tex]\( x \)[/tex] approaches [tex]\(0\)[/tex] from both positive and negative sides, [tex]\(\frac{1}{x}\)[/tex] approaches [tex]\(\pm \infty\)[/tex], making [tex]\(\frac{1}{x^2}\)[/tex] approach [tex]\(\infty\)[/tex], causing [tex]\(-\frac{2}{x^2}\)[/tex] to approach [tex]\(-\infty\)[/tex]. Thus, [tex]\(-\frac{2}{x^2} + 3\)[/tex] approaches [tex]\(-\infty\)[/tex].

5. Combine the observations to form the range:
- The function approaches the upper bound of [tex]\(3\)[/tex] but never reaches it as [tex]\(x\)[/tex] heads towards [tex]\(\infty\)[/tex] or [tex]\(-\infty\)[/tex].
- The function decreases without bound as [tex]\(x\)[/tex] approaches [tex]\(0\)[/tex].

Therefore, combining these observations, the range of [tex]\((u \circ v)(x)\)[/tex] is:
[tex]\[ (-\infty, 3) \][/tex]

The correct answer is:
[tex]\[(-\infty, 3)\][/tex]
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