Algebra Examples

$f (x) = - x^{2} - 1$ , $g (x) = x + 5$

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

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

$f (x + 5) = - {(x + 5)}^{2} - 1$

Step 3

Simplify each term.

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Step 3.1

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

$f (x + 5) = - ((x + 5) (x + 5)) - 1$

Step 3.2

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

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Step 3.2.1

Apply the distributive property.

$f (x + 5) = - (x (x + 5) + 5 (x + 5)) - 1$

Step 3.2.2

Apply the distributive property.

$f (x + 5) = - (x \cdot x + x \cdot 5 + 5 (x + 5)) - 1$

Step 3.2.3

Apply the distributive property.

$f (x + 5) = - (x \cdot x + x \cdot 5 + 5 x + 5 \cdot 5) - 1$

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 $x$ .

$f (x + 5) = - (x^{2} + x \cdot 5 + 5 x + 5 \cdot 5) - 1$

Step 3.3.1.2

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

$f (x + 5) = - (x^{2} + 5 \cdot x + 5 x + 5 \cdot 5) - 1$

Step 3.3.1.3

Multiply $5$ by $5$ .

$f (x + 5) = - (x^{2} + 5 x + 5 x + 25) - 1$

Step 3.3.2

Add $5 x$ and $5 x$ .

$f (x + 5) = - (x^{2} + 10 x + 25) - 1$

Step 3.4

Apply the distributive property.

$f (x + 5) = - x^{2} - (10 x) - 1 \cdot 25 - 1$

Step 3.5

Simplify.

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Step 3.5.1

Multiply $10$ by $- 1$ .

$f (x + 5) = - x^{2} - 10 x - 1 \cdot 25 - 1$

Step 3.5.2

Multiply $- 1$ by $25$ .

$f (x + 5) = - x^{2} - 10 x - 25 - 1$

Step 4

Subtract $1$ from $- 25$ .

$f (x + 5) = - x^{2} - 10 x - 26$