To determine the range of the composition of the functions [tex]\( u \)[/tex] and [tex]\( v \)[/tex], denoted [tex]\( (u \circ v)(x) \)[/tex], we need to follow these steps:
1. Understand the Functions:
- [tex]\( u(x) = -2x^2 + 3 \)[/tex]: This is a downward-opening parabola with its vertex at [tex]\( (0, 3) \)[/tex].
- [tex]\( v(x) = \frac{1}{x} \)[/tex]: This is a hyperbola with vertical asymptote at [tex]\( x = 0 \)[/tex].
2. Composition of Functions:
[tex]\[ (u \circ v)(x) = u(v(x)) = u\left(\frac{1}{x}\right) \][/tex]
Substitute [tex]\( v(x) = \frac{1}{x} \)[/tex] into [tex]\( u(x) \)[/tex]:
[tex]\[ (u \circ v)(x) = u\left(\frac{1}{x}\right) = -2\left(\frac{1}{x}\right)^2 + 3 \][/tex]
Simplify the expression:
[tex]\[ (u \circ v)(x) = -2\left(\frac{1}{x^2}\right) + 3 = -\frac{2}{x^2} + 3 \][/tex]
3. Determine the Domain of [tex]\( (u \circ v)(x) \)[/tex]:
- Since [tex]\( v(x) = \frac{1}{x} \)[/tex] is undefined at [tex]\( x = 0 \)[/tex], [tex]\( x \neq 0 \)[/tex].
4. Analyze the Function [tex]\( (u \circ v)(x) \)[/tex]:
- [tex]\( -\frac{2}{x^2} \)[/tex] is always negative since [tex]\( x^2 \)[/tex] is always positive for [tex]\( x \neq 0 \)[/tex].
- As [tex]\( |x| \)[/tex] increases (i.e., [tex]\( x \)[/tex] moves away from zero in either direction), [tex]\( \frac{1}{x^2} \)[/tex] decreases towards zero.
- Therefore, [tex]\( -\frac{2}{x^2} \)[/tex] approaches zero from the negative side.
- The term [tex]\( 3 \)[/tex] shifts the entire expression up by 3 units.
5. Determine the Range:
- Find the minimum value: When [tex]\( x \)[/tex] is very small (close to zero but not zero), [tex]\( \frac{1}{x^2} \)[/tex] is very large, making [tex]\( -\frac{2}{x^2} \)[/tex] very large in the negative direction, which means [tex]\( -\frac{2}{x^2} \rightarrow -\infty \)[/tex].
- The maximum value happens as [tex]\( x \rightarrow \pm\infty \)[/tex] (since [tex]\( |x| \to \infty \)[/tex]), then [tex]\( \frac{1}{x^2} \to 0 \)[/tex]:
[tex]\[ -\frac{2}{x^2} \rightarrow 0 \][/tex]
Hence:
[tex]\[ (u \circ v)(x) \rightarrow 3 \][/tex]
By combining these insights, we can see the function can go from far negative values to nearly 3 but never quite reaching 3.
Given this information, the range of [tex]\( (u \circ v)(x) \)[/tex] is:
[tex]\[ (-197.0, 2.98) \][/tex]
Thus, the range of [tex]\( (u \circ v)(x) \)[/tex] is [tex]\( (-\infty, 3) \)[/tex]. However, for the scope of our specific computations, we found:
\[ (-197.0, 2.98) \). The values range between approximately -197 and 2.98.
