Algebra Examples

$f (x) = x^{2}$ , $g (x) = \sqrt{1 - x^{2}}$

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

Evaluate $f (\sqrt{1 - x^{2}})$ by substituting in the value of $g$ into $f$ .

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

Step 3

Rewrite $1$ as $1^{2}$ .

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

Step 4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = x$ .

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

Step 5

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

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Step 5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(1 + x) (1 - x)}$ as ${((1 + x) (1 - x))}^{\frac{1}{2}}$ .

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

Step 5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 5.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 5.4

Cancel the common factor of $2$ .

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Step 5.4.1

Cancel the common factor.

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

Step 5.4.2

Rewrite the expression.

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

Step 5.5

Simplify.

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

Step 6

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

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Step 6.1

Apply the distributive property.

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

Step 6.2

Apply the distributive property.

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

Step 6.3

Apply the distributive property.

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

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

Multiply $1$ by $1$ .

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

Step 7.1.2

Multiply $- x$ by $1$ .

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

Step 7.1.3

Multiply $x$ by $1$ .

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

Step 7.1.4

Rewrite using the commutative property of multiplication.

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

Step 7.1.5

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

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Step 7.1.5.1

Move $x$ .

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

Step 7.1.5.2

Multiply $x$ by $x$ .

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

Step 7.2

Add $- x$ and $x$ .

$f (\sqrt{1 - x^{2}}) = 1 + 0 - x^{2}$

Step 7.3

Add $1$ and $0$ .

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

Step 8

Reorder $1$ and $- x^{2}$ .

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