Algebra Examples

$f (x) = \frac{1}{23 x - 1}$ $g (x) = \frac{1}{x^{2}}$

Step 1

Set up the composite result function.

$g (f (x))$

Step 2

Evaluate $g (\frac{1}{23 x - 1})$ by substituting in the value of $f$ into $g$ .

$g (\frac{1}{23 x - 1}) = \frac{1}{{(\frac{1}{23 x - 1})}^{2}}$

Step 3

Simplify the denominator.

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Step 3.1

Apply the product rule to $\frac{1}{23 x - 1}$ .

$g (\frac{1}{23 x - 1}) = \frac{1}{\frac{1^{2}}{{(23 x - 1)}^{2}}}$

Step 3.2

One to any power is one.

$g (\frac{1}{23 x - 1}) = \frac{1}{\frac{1}{{(23 x - 1)}^{2}}}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$g (\frac{1}{23 x - 1}) = 1 {(23 x - 1)}^{2}$

Step 5

Simplify the expression.

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Step 5.1

Multiply ${(23 x - 1)}^{2}$ by $1$ .

$g (\frac{1}{23 x - 1}) = {(23 x - 1)}^{2}$

Step 5.2

Rewrite ${(23 x - 1)}^{2}$ as $(23 x - 1) (23 x - 1)$ .

$g (\frac{1}{23 x - 1}) = (23 x - 1) (23 x - 1)$

Step 6

Expand $(23 x - 1) (23 x - 1)$ using the FOIL Method.

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Step 6.1

Apply the distributive property.

$g (\frac{1}{23 x - 1}) = 23 x (23 x - 1) - 1 (23 x - 1)$

Step 6.2

Apply the distributive property.

$g (\frac{1}{23 x - 1}) = 23 x (23 x) + 23 x \cdot - 1 - 1 (23 x - 1)$

Step 6.3

Apply the distributive property.

$g (\frac{1}{23 x - 1}) = 23 x (23 x) + 23 x \cdot - 1 - 1 (23 x) - 1 \cdot - 1$

Step 7

Simplify and combine like terms.

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Step 7.1

Simplify each term.

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Step 7.1.1

Rewrite using the commutative property of multiplication.

$g (\frac{1}{23 x - 1}) = 23 \cdot (23 x \cdot x) + 23 x \cdot - 1 - 1 (23 x) - 1 \cdot - 1$

Step 7.1.2

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

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Step 7.1.2.1

Move $x$ .

$g (\frac{1}{23 x - 1}) = 23 \cdot (23 (x \cdot x)) + 23 x \cdot - 1 - 1 (23 x) - 1 \cdot - 1$

Step 7.1.2.2

Multiply $x$ by $x$ .

$g (\frac{1}{23 x - 1}) = 23 \cdot (23 x^{2}) + 23 x \cdot - 1 - 1 (23 x) - 1 \cdot - 1$

Step 7.1.3

Multiply $23$ by $23$ .

$g (\frac{1}{23 x - 1}) = 529 x^{2} + 23 x \cdot - 1 - 1 (23 x) - 1 \cdot - 1$

Step 7.1.4

Multiply $- 1$ by $23$ .

$g (\frac{1}{23 x - 1}) = 529 x^{2} - 23 x - 1 (23 x) - 1 \cdot - 1$

Step 7.1.5

Multiply $23$ by $- 1$ .

$g (\frac{1}{23 x - 1}) = 529 x^{2} - 23 x - 23 x - 1 \cdot - 1$

Step 7.1.6

Multiply $- 1$ by $- 1$ .

$g (\frac{1}{23 x - 1}) = 529 x^{2} - 23 x - 23 x + 1$

Step 7.2

Subtract $23 x$ from $- 23 x$ .

$g (\frac{1}{23 x - 1}) = 529 x^{2} - 46 x + 1$