Find f(g(x)) f(x)=x^2 , g(x) = square root of 1-x^2

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

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 $(1 + x) (1 - x)$ .

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Step 6

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

Apply the distributive property.

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

Apply the distributive property.

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

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.

Simplify each term.

Multiply $1$ by $1$ .

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

Multiply $- x$ by $1$ .

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

Multiply $x$ by $1$ .

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

Rewrite using the commutative property of multiplication.

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Add $- x$ and $x$ .

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

Add $1$ and $0$ .

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

Step 8

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

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

Find f(g(x)) f(x)=x^2 , g(x) = square root of 1-x^2

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