Precalculus Examples

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

Step 1

Set up the composite result function.

$u (v (x))$

Step 2

Evaluate $u (\frac{1}{x})$ by substituting in the value of $v$ into $u$ .

$u (\frac{1}{x}) = - 2 {(\frac{1}{x})}^{2} + 3$

Step 3

Simplify each term.

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Step 3.1

Apply the product rule to $\frac{1}{x}$ .

$u (\frac{1}{x}) = - 2 \frac{1^{2}}{x^{2}} + 3$

Step 3.2

One to any power is one.

$u (\frac{1}{x}) = - 2 \frac{1}{x^{2}} + 3$

Step 3.3

Combine $- 2$ and $\frac{1}{x^{2}}$ .

$u (\frac{1}{x}) = \frac{- 2}{x^{2}} + 3$

Step 3.4

Move the negative in front of the fraction.

$u (\frac{1}{x}) = - \frac{2}{x^{2}} + 3$