Algebra Examples

$f (x) = \frac{x - 1}{x^{2} + x - 1}$ $g (x) = 3^{x} - 2$

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

Evaluate $f (3^{x} - 2)$ by substituting in the value of $g$ into $f$ .

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

Step 3

Simplify the expression.

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Step 3.1

Remove parentheses.

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

Step 3.2

Subtract $1$ from $- 2$ .

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

Step 4

Simplify the denominator.

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Step 4.1

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

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

Step 4.2

Expand $(3^{x} - 2) (3^{x} - 2)$ using the FOIL Method.

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Step 4.2.1

Apply the distributive property.

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

Step 4.2.2

Apply the distributive property.

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

Step 4.2.3

Apply the distributive property.

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

Step 4.3

Simplify and combine like terms.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Multiply $3^{x}$ by $3^{x}$ by adding the exponents.

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Step 4.3.1.1.1

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

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

Step 4.3.1.1.2

Add $x$ and $x$ .

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

Step 4.3.1.2

Move $- 2$ to the left of $3^{x}$ .

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

Step 4.3.1.3

Multiply $- 2$ by $- 2$ .

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

Step 4.3.2

Subtract $2 \cdot 3^{x}$ from $- 2 \cdot 3^{x}$ .

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

Step 4.4

Add $- 4 \cdot 3^{x}$ and $3^{x}$ .

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

Step 4.5

Subtract $2$ from $4$ .

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

Step 4.6

Subtract $1$ from $2$ .

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

Step 4.7

Multiply $- 3 \cdot 3^{x}$ .

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Step 4.7.1

Factor out negative.

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

Step 4.7.2

Raise $3$ to the power of $1$ .

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

Step 4.7.3

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

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