Algebra Examples

$g (x) = \frac{x}{4 - x}$ , $h (x) = x^{3} - 2 x^{2} + 4$

Step 1

Set up the composite result function.

$g (h (x))$

Step 2

Evaluate $g (x^{3} - 2 x^{2} + 4)$ by substituting in the value of $h$ into $g$ .

$g (x^{3} - 2 x^{2} + 4) = \frac{x^{3} - 2 x^{2} + 4}{4 - (x^{3} - 2 x^{2} + 4)}$

Step 3

Simplify the denominator.

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Step 3.1

Apply the distributive property.

$g (x^{3} - 2 x^{2} + 4) = \frac{x^{3} - 2 x^{2} + 4}{4 - x^{3} - (- 2 x^{2}) - 1 \cdot 4}$

Step 3.2

Simplify.

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Step 3.2.1

Multiply $- 2$ by $- 1$ .

$g (x^{3} - 2 x^{2} + 4) = \frac{x^{3} - 2 x^{2} + 4}{4 - x^{3} + 2 x^{2} - 1 \cdot 4}$

Step 3.2.2

Multiply $- 1$ by $4$ .

$g (x^{3} - 2 x^{2} + 4) = \frac{x^{3} - 2 x^{2} + 4}{4 - x^{3} + 2 x^{2} - 4}$

Step 3.3

Subtract $4$ from $4$ .

$g (x^{3} - 2 x^{2} + 4) = \frac{x^{3} - 2 x^{2} + 4}{- x^{3} + 2 x^{2} + 0}$

Step 3.4

Add $- x^{3} + 2 x^{2}$ and $0$ .

$g (x^{3} - 2 x^{2} + 4) = \frac{x^{3} - 2 x^{2} + 4}{- x^{3} + 2 x^{2}}$

Step 3.5

Factor $x^{2}$ out of $- x^{3} + 2 x^{2}$ .

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Step 3.5.1

Factor $x^{2}$ out of $- x^{3}$ .

$g (x^{3} - 2 x^{2} + 4) = \frac{x^{3} - 2 x^{2} + 4}{x^{2} (- x) + 2 x^{2}}$

Step 3.5.2

Factor $x^{2}$ out of $2 x^{2}$ .

$g (x^{3} - 2 x^{2} + 4) = \frac{x^{3} - 2 x^{2} + 4}{x^{2} (- x) + x^{2} \cdot 2}$

Step 3.5.3

Factor $x^{2}$ out of $x^{2} (- x) + x^{2} \cdot 2$ .

$g (x^{3} - 2 x^{2} + 4) = \frac{x^{3} - 2 x^{2} + 4}{x^{2} (- x + 2)}$

Step 4

Simplify with factoring out.

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Step 4.1

Factor $- 1$ out of $- x$ .

$g (x^{3} - 2 x^{2} + 4) = \frac{x^{3} - 2 x^{2} + 4}{x^{2} (- (x) + 2)}$

Step 4.2

Rewrite $2$ as $- 1 (- 2)$ .

$g (x^{3} - 2 x^{2} + 4) = \frac{x^{3} - 2 x^{2} + 4}{x^{2} (- (x) - 1 \cdot - 2)}$

Step 4.3

Factor $- 1$ out of $- (x) - 1 (- 2)$ .

$g (x^{3} - 2 x^{2} + 4) = \frac{x^{3} - 2 x^{2} + 4}{x^{2} (- (x - 2))}$

Step 4.4

Rewrite negatives.

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Step 4.4.1

Rewrite $- (x - 2)$ as $- 1 (x - 2)$ .

$g (x^{3} - 2 x^{2} + 4) = \frac{x^{3} - 2 x^{2} + 4}{x^{2} (- 1 (x - 2))}$

Step 4.4.2

Move the negative in front of the fraction.

$g (x^{3} - 2 x^{2} + 4) = - \frac{x^{3} - 2 x^{2} + 4}{x^{2} (x - 2)}$