Algebra Examples

$f (x) = \frac{x - 8}{x}$ , $g (x) = x^{2} + 6$

Step 1

Set up the composite result function.

$g (f (x))$

Step 2

Evaluate $g (\frac{x - 8}{x})$ by substituting in the value of $f$ into $g$ .

$g (\frac{x - 8}{x}) = {(\frac{x - 8}{x})}^{2} + 6$

Step 3

Apply the product rule to $\frac{x - 8}{x}$ .

$g (\frac{x - 8}{x}) = \frac{{(x - 8)}^{2}}{x^{2}} + 6$

Step 4

To write $6$ as a fraction with a common denominator, multiply by $\frac{x^{2}}{x^{2}}$ .

$g (\frac{x - 8}{x}) = \frac{{(x - 8)}^{2}}{x^{2}} + \frac{6 x^{2}}{x^{2}}$

Step 5

Combine the numerators over the common denominator.

$g (\frac{x - 8}{x}) = \frac{{(x - 8)}^{2} + 6 x^{2}}{x^{2}}$

Step 6

Simplify the numerator.

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Step 6.1

Rewrite ${(x - 8)}^{2}$ as $(x - 8) (x - 8)$ .

$g (\frac{x - 8}{x}) = \frac{(x - 8) (x - 8) + 6 x^{2}}{x^{2}}$

Step 6.2

Expand $(x - 8) (x - 8)$ using the FOIL Method.

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Step 6.2.1

Apply the distributive property.

$g (\frac{x - 8}{x}) = \frac{x (x - 8) - 8 (x - 8) + 6 x^{2}}{x^{2}}$

Step 6.2.2

Apply the distributive property.

$g (\frac{x - 8}{x}) = \frac{x \cdot x + x \cdot - 8 - 8 (x - 8) + 6 x^{2}}{x^{2}}$

Step 6.2.3

Apply the distributive property.

$g (\frac{x - 8}{x}) = \frac{x \cdot x + x \cdot - 8 - 8 x - 8 \cdot - 8 + 6 x^{2}}{x^{2}}$

Step 6.3

Simplify and combine like terms.

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Step 6.3.1

Simplify each term.

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Step 6.3.1.1

Multiply $x$ by $x$ .

$g (\frac{x - 8}{x}) = \frac{x^{2} + x \cdot - 8 - 8 x - 8 \cdot - 8 + 6 x^{2}}{x^{2}}$

Step 6.3.1.2

Move $- 8$ to the left of $x$ .

$g (\frac{x - 8}{x}) = \frac{x^{2} - 8 \cdot x - 8 x - 8 \cdot - 8 + 6 x^{2}}{x^{2}}$

Step 6.3.1.3

Multiply $- 8$ by $- 8$ .

$g (\frac{x - 8}{x}) = \frac{x^{2} - 8 x - 8 x + 64 + 6 x^{2}}{x^{2}}$

Step 6.3.2

Subtract $8 x$ from $- 8 x$ .

$g (\frac{x - 8}{x}) = \frac{x^{2} - 16 x + 64 + 6 x^{2}}{x^{2}}$

Step 6.4

Add $x^{2}$ and $6 x^{2}$ .

$g (\frac{x - 8}{x}) = \frac{7 x^{2} - 16 x + 64}{x^{2}}$