Algebra Examples

$f (x) = x^{2} - 9$ , $g (x) = 3 - x$

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

Evaluate $f (3 - x)$ by substituting in the value of $g$ into $f$ .

$f (3 - x) = {(3 - x)}^{2} - 9$

Step 3

Simplify each term.

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Step 3.1

Rewrite ${(3 - x)}^{2}$ as $(3 - x) (3 - x)$ .

$f (3 - x) = (3 - x) (3 - x) - 9$

Step 3.2

Expand $(3 - x) (3 - x)$ using the FOIL Method.

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Step 3.2.1

Apply the distributive property.

$f (3 - x) = 3 (3 - x) - x (3 - x) - 9$

Step 3.2.2

Apply the distributive property.

$f (3 - x) = 3 \cdot 3 + 3 (- x) - x (3 - x) - 9$

Step 3.2.3

Apply the distributive property.

$f (3 - x) = 3 \cdot 3 + 3 (- x) - x \cdot 3 - x (- x) - 9$

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 $3$ by $3$ .

$f (3 - x) = 9 + 3 (- x) - x \cdot 3 - x (- x) - 9$

Step 3.3.1.2

Multiply $- 1$ by $3$ .

$f (3 - x) = 9 - 3 x - x \cdot 3 - x (- x) - 9$

Step 3.3.1.3

Multiply $3$ by $- 1$ .

$f (3 - x) = 9 - 3 x - 3 x - x (- x) - 9$

Step 3.3.1.4

Rewrite using the commutative property of multiplication.

$f (3 - x) = 9 - 3 x - 3 x - 1 \cdot (- 1 x \cdot x) - 9$

Step 3.3.1.5

Multiply $x$ by $x$ by adding the exponents.

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Step 3.3.1.5.1

Move $x$ .

$f (3 - x) = 9 - 3 x - 3 x - 1 \cdot (- 1 (x \cdot x)) - 9$

Step 3.3.1.5.2

Multiply $x$ by $x$ .

$f (3 - x) = 9 - 3 x - 3 x - 1 \cdot (- 1 x^{2}) - 9$

Step 3.3.1.6

Multiply $- 1$ by $- 1$ .

$f (3 - x) = 9 - 3 x - 3 x + 1 x^{2} - 9$

Step 3.3.1.7

Multiply $x^{2}$ by $1$ .

$f (3 - x) = 9 - 3 x - 3 x + x^{2} - 9$

Step 3.3.2

Subtract $3 x$ from $- 3 x$ .

$f (3 - x) = 9 - 6 x + x^{2} - 9$

Step 4

Simplify by adding terms.

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Step 4.1

Combine the opposite terms in $9 - 6 x + x^{2} - 9$ .

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Step 4.1.1

Subtract $9$ from $9$ .

$f (3 - x) = - 6 x + x^{2} + 0$

Step 4.1.2

Add $- 6 x + x^{2}$ and $0$ .

$f (3 - x) = - 6 x + x^{2}$

Step 4.2

Reorder $- 6 x$ and $x^{2}$ .

$f (3 - x) = x^{2} - 6 x$