Precalculus Examples

$f (x) = x^{2} + 3 x + 4$ , $g (x) = x - 1$ , fog

Set up the composite result function.

$f (g (x))$

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

${(x - 1)}^{2} + 3 (x - 1) + 4$

Simplify each term.

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Rewrite ${(x - 1)}^{2}$ as $(x - 1) (x - 1)$ .

$f (x - 1) = (x - 1) (x - 1) + 3 (x - 1) + 4$

Expand $(x - 1) (x - 1)$ using the FOIL Method.

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Apply the distributive property.

$f (x - 1) = x (x - 1) - 1 (x - 1) + 3 (x - 1) + 4$

Apply the distributive property.

$f (x - 1) = x x + x \cdot - 1 - 1 (x - 1) + 3 (x - 1) + 4$

Apply the distributive property.

$f (x - 1) = x x + x \cdot - 1 + (- 1 x - 1 \cdot - 1) + 3 (x - 1) + 4$

Remove parentheses.

$f (x - 1) = x x + x \cdot - 1 - 1 x - 1 \cdot - 1 + 3 (x - 1) + 4$

Simplify and combine like terms.

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Simplify each term.

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Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (x - 1) = x^{1 + 1} + x \cdot - 1 - 1 x - 1 \cdot - 1 + 3 (x - 1) + 4$

Add $1$ and $1$ to get $2$ .

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

Move $- 1$ to the left of the expression $x \cdot - 1$ .

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

Multiply $- 1$ by $x$ to get $- 1 x$ .

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

Rewrite $- 1 x$ as $- x$ .

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

Rewrite $- 1 x$ as $- x$ .

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

Multiply $- 1$ by $- 1$ to get $1$ .

$f (x - 1) = x^{2} - x - x + 1 + 3 (x - 1) + 4$

Subtract $x$ from $- x$ to get $- 2 x$ .

$f (x - 1) = x^{2} - 2 x + 1 + 3 (x - 1) + 4$

Apply the distributive property.

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

Multiply $3$ by $- 1$ to get $- 3$ .

$f (x - 1) = x^{2} - 2 x + 1 + (3 x - 3) + 4$

Simplify by adding terms.

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Remove unnecessary parentheses.

$f (x - 1) = x^{2} - 2 x + 1 + 3 x - 3 + 4$

Add $- 2 x$ and $3 x$ to get $x$ .

$f (x - 1) = x^{2} + x + 1 - 3 + 4$

Simplify by adding and subtracting.

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Subtract $3$ from $1$ to get $- 2$ .

$f (x - 1) = x^{2} + x - 2 + 4$

Add $- 2$ and $4$ to get $2$ .

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

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