Algebra Examples

$f (x) = \frac{x - 5}{3}$ , $g (x) = x^{2} + 5$

Step 1

Set up the composite result function.

$g (f (x))$

Step 2

Evaluate $g (\frac{x - 5}{3})$ by substituting in the value of $f$ into $g$ .

$g (\frac{x - 5}{3}) = {(\frac{x - 5}{3})}^{2} + 5$

Step 3

Simplify each term.

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Step 3.1

Apply the product rule to $\frac{x - 5}{3}$ .

$g (\frac{x - 5}{3}) = \frac{{(x - 5)}^{2}}{3^{2}} + 5$

Step 3.2

Raise $3$ to the power of $2$ .

$g (\frac{x - 5}{3}) = \frac{{(x - 5)}^{2}}{9} + 5$

Step 4

To write $5$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$g (\frac{x - 5}{3}) = \frac{{(x - 5)}^{2}}{9} + 5 \cdot \frac{9}{9}$

Step 5

Simplify terms.

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Step 5.1

Combine $5$ and $\frac{9}{9}$ .

$g (\frac{x - 5}{3}) = \frac{{(x - 5)}^{2}}{9} + \frac{5 \cdot 9}{9}$

Step 5.2

Combine the numerators over the common denominator.

$g (\frac{x - 5}{3}) = \frac{{(x - 5)}^{2} + 5 \cdot 9}{9}$

Step 6

Simplify the numerator.

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Step 6.1

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

$g (\frac{x - 5}{3}) = \frac{(x - 5) (x - 5) + 5 \cdot 9}{9}$

Step 6.2

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

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Step 6.2.1

Apply the distributive property.

$g (\frac{x - 5}{3}) = \frac{x (x - 5) - 5 (x - 5) + 5 \cdot 9}{9}$

Step 6.2.2

Apply the distributive property.

$g (\frac{x - 5}{3}) = \frac{x \cdot x + x \cdot - 5 - 5 (x - 5) + 5 \cdot 9}{9}$

Step 6.2.3

Apply the distributive property.

$g (\frac{x - 5}{3}) = \frac{x \cdot x + x \cdot - 5 - 5 x - 5 \cdot - 5 + 5 \cdot 9}{9}$

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 $x$ .

$g (\frac{x - 5}{3}) = \frac{x^{2} + x \cdot - 5 - 5 x - 5 \cdot - 5 + 5 \cdot 9}{9}$

Step 6.3.1.2

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

$g (\frac{x - 5}{3}) = \frac{x^{2} - 5 \cdot x - 5 x - 5 \cdot - 5 + 5 \cdot 9}{9}$

Step 6.3.1.3

Multiply $- 5$ by $- 5$ .

$g (\frac{x - 5}{3}) = \frac{x^{2} - 5 x - 5 x + 25 + 5 \cdot 9}{9}$

Step 6.3.2

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

$g (\frac{x - 5}{3}) = \frac{x^{2} - 10 x + 25 + 5 \cdot 9}{9}$

Step 6.4

Multiply $5$ by $9$ .

$g (\frac{x - 5}{3}) = \frac{x^{2} - 10 x + 25 + 45}{9}$

Step 6.5

Add $25$ and $45$ .

$g (\frac{x - 5}{3}) = \frac{x^{2} - 10 x + 70}{9}$