Algebra Examples

$f (x) = x - 5$ $g (x) = \frac{x^{2} - 12}{x^{2} + 4}$

Step 1

Set up the composite result function.

$g (f (x))$

Step 2

Evaluate $g (x - 5)$ by substituting in the value of $f$ into $g$ .

$g (x - 5) = \frac{{(x - 5)}^{2} - 12}{{(x - 5)}^{2} + 4}$

Step 3

Simplify the numerator.

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Step 3.1

Rewrite ${(x - 5)}^{2}$ as $(x - 5) (x - 5)$ .

$g (x - 5) = \frac{(x - 5) (x - 5) - 12}{{(x - 5)}^{2} + 4}$

Step 3.2

Expand $(x - 5) (x - 5)$ using the FOIL Method.

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Step 3.2.1

Apply the distributive property.

$g (x - 5) = \frac{x (x - 5) - 5 (x - 5) - 12}{{(x - 5)}^{2} + 4}$

Step 3.2.2

Apply the distributive property.

$g (x - 5) = \frac{x \cdot x + x \cdot - 5 - 5 (x - 5) - 12}{{(x - 5)}^{2} + 4}$

Step 3.2.3

Apply the distributive property.

$g (x - 5) = \frac{x \cdot x + x \cdot - 5 - 5 x - 5 \cdot - 5 - 12}{{(x - 5)}^{2} + 4}$

Step 3.3

Simplify and combine like terms.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Multiply $x$ by $x$ .

$g (x - 5) = \frac{x^{2} + x \cdot - 5 - 5 x - 5 \cdot - 5 - 12}{{(x - 5)}^{2} + 4}$

Step 3.3.1.2

Move $- 5$ to the left of $x$ .

$g (x - 5) = \frac{x^{2} - 5 \cdot x - 5 x - 5 \cdot - 5 - 12}{{(x - 5)}^{2} + 4}$

Step 3.3.1.3

Multiply $- 5$ by $- 5$ .

$g (x - 5) = \frac{x^{2} - 5 x - 5 x + 25 - 12}{{(x - 5)}^{2} + 4}$

Step 3.3.2

Subtract $5 x$ from $- 5 x$ .

$g (x - 5) = \frac{x^{2} - 10 x + 25 - 12}{{(x - 5)}^{2} + 4}$

Step 3.4

Subtract $12$ from $25$ .

$g (x - 5) = \frac{x^{2} - 10 x + 13}{{(x - 5)}^{2} + 4}$

Step 4

Simplify the denominator.

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Step 4.1

Rewrite ${(x - 5)}^{2}$ as $(x - 5) (x - 5)$ .

$g (x - 5) = \frac{x^{2} - 10 x + 13}{(x - 5) (x - 5) + 4}$

Step 4.2

Expand $(x - 5) (x - 5)$ using the FOIL Method.

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Step 4.2.1

Apply the distributive property.

$g (x - 5) = \frac{x^{2} - 10 x + 13}{x (x - 5) - 5 (x - 5) + 4}$

Step 4.2.2

Apply the distributive property.

$g (x - 5) = \frac{x^{2} - 10 x + 13}{x \cdot x + x \cdot - 5 - 5 (x - 5) + 4}$

Step 4.2.3

Apply the distributive property.

$g (x - 5) = \frac{x^{2} - 10 x + 13}{x \cdot x + x \cdot - 5 - 5 x - 5 \cdot - 5 + 4}$

Step 4.3

Simplify and combine like terms.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Multiply $x$ by $x$ .

$g (x - 5) = \frac{x^{2} - 10 x + 13}{x^{2} + x \cdot - 5 - 5 x - 5 \cdot - 5 + 4}$

Step 4.3.1.2

Move $- 5$ to the left of $x$ .

$g (x - 5) = \frac{x^{2} - 10 x + 13}{x^{2} - 5 \cdot x - 5 x - 5 \cdot - 5 + 4}$

Step 4.3.1.3

Multiply $- 5$ by $- 5$ .

$g (x - 5) = \frac{x^{2} - 10 x + 13}{x^{2} - 5 x - 5 x + 25 + 4}$

Step 4.3.2

Subtract $5 x$ from $- 5 x$ .

$g (x - 5) = \frac{x^{2} - 10 x + 13}{x^{2} - 10 x + 25 + 4}$

Step 4.4

Add $25$ and $4$ .

$g (x - 5) = \frac{x^{2} - 10 x + 13}{x^{2} - 10 x + 29}$