Trigonometry Examples

$y = \sqrt{4 - x} + 1$

Step 1

Interchange the variables.

$x = \sqrt{4 - y} + 1$

Step 2

Solve for $y$ .

Tap for more steps...

Step 2.1

Rewrite the equation as $\sqrt{4 - y} + 1 = x$ .

$\sqrt{4 - y} + 1 = x$

Step 2.2

Subtract $1$ from both sides of the equation.

$\sqrt{4 - y} = x - 1$

Step 2.3

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{4 - y}}^{2} = {(x - 1)}^{2}$

Step 2.4

Simplify each side of the equation.

Tap for more steps...

Step 2.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 - y}$ as ${(4 - y)}^{\frac{1}{2}}$ .

${({(4 - y)}^{\frac{1}{2}})}^{2} = {(x - 1)}^{2}$

Step 2.4.2

Simplify the left side.

Tap for more steps...

Step 2.4.2.1

Simplify ${({(4 - y)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 2.4.2.1.1

Multiply the exponents in ${({(4 - y)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 2.4.2.1.1.1

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

${(4 - y)}^{\frac{1}{2} \cdot 2} = {(x - 1)}^{2}$

Step 2.4.2.1.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.4.2.1.1.2.1

Cancel the common factor.

${(4 - y)}^{\frac{1}{2} \cdot 2} = {(x - 1)}^{2}$

Step 2.4.2.1.1.2.2

Rewrite the expression.

${(4 - y)}^{1} = {(x - 1)}^{2}$

Step 2.4.2.1.2

Simplify.

$4 - y = {(x - 1)}^{2}$

Step 2.4.3

Simplify the right side.

Tap for more steps...

Step 2.4.3.1

Simplify ${(x - 1)}^{2}$ .

Tap for more steps...

Step 2.4.3.1.1

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

$4 - y = (x - 1) (x - 1)$

Step 2.4.3.1.2

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

Tap for more steps...

Step 2.4.3.1.2.1

Apply the distributive property.

$4 - y = x (x - 1) - 1 (x - 1)$

Step 2.4.3.1.2.2

Apply the distributive property.

$4 - y = x \cdot x + x \cdot - 1 - 1 (x - 1)$

Step 2.4.3.1.2.3

Apply the distributive property.

$4 - y = x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1$

Step 2.4.3.1.3

Simplify and combine like terms.

Tap for more steps...

Step 2.4.3.1.3.1

Simplify each term.

Tap for more steps...

Step 2.4.3.1.3.1.1

Multiply $x$ by $x$ .

$4 - y = x^{2} + x \cdot - 1 - 1 x - 1 \cdot - 1$

Step 2.4.3.1.3.1.2

Move $- 1$ to the left of $x$ .

$4 - y = x^{2} - 1 \cdot x - 1 x - 1 \cdot - 1$

Step 2.4.3.1.3.1.3

Rewrite $- 1 x$ as $- x$ .

$4 - y = x^{2} - x - 1 x - 1 \cdot - 1$

Step 2.4.3.1.3.1.4

Rewrite $- 1 x$ as $- x$ .

$4 - y = x^{2} - x - x - 1 \cdot - 1$

Step 2.4.3.1.3.1.5

Multiply $- 1$ by $- 1$ .

$4 - y = x^{2} - x - x + 1$

Step 2.4.3.1.3.2

Subtract $x$ from $- x$ .

$4 - y = x^{2} - 2 x + 1$

Step 2.5

Solve for $y$ .

Tap for more steps...

Step 2.5.1

Move all terms not containing $y$ to the right side of the equation.

Tap for more steps...

Step 2.5.1.1

Subtract $4$ from both sides of the equation.

$- y = x^{2} - 2 x + 1 - 4$

Step 2.5.1.2

Subtract $4$ from $1$ .

$- y = x^{2} - 2 x - 3$

Step 2.5.2

Divide each term in $- y = x^{2} - 2 x - 3$ by $- 1$ and simplify.

Tap for more steps...

Step 2.5.2.1

Divide each term in $- y = x^{2} - 2 x - 3$ by $- 1$ .

$\frac{- y}{- 1} = \frac{x^{2}}{- 1} + \frac{- 2 x}{- 1} + \frac{- 3}{- 1}$

Step 2.5.2.2

Simplify the left side.

Tap for more steps...

Step 2.5.2.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} = \frac{x^{2}}{- 1} + \frac{- 2 x}{- 1} + \frac{- 3}{- 1}$

Step 2.5.2.2.2

Divide $y$ by $1$ .

$y = \frac{x^{2}}{- 1} + \frac{- 2 x}{- 1} + \frac{- 3}{- 1}$

Step 2.5.2.3

Simplify the right side.

Tap for more steps...

Step 2.5.2.3.1

Simplify each term.

Tap for more steps...

Step 2.5.2.3.1.1

Move the negative one from the denominator of $\frac{x^{2}}{- 1}$ .

$y = - 1 \cdot x^{2} + \frac{- 2 x}{- 1} + \frac{- 3}{- 1}$

Step 2.5.2.3.1.2

Rewrite $- 1 \cdot x^{2}$ as $- x^{2}$ .

$y = - x^{2} + \frac{- 2 x}{- 1} + \frac{- 3}{- 1}$

Step 2.5.2.3.1.3

Move the negative one from the denominator of $\frac{- 2 x}{- 1}$ .

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

Step 2.5.2.3.1.4

Rewrite $- 1 \cdot (- 2 x)$ as $- (- 2 x)$ .

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

Step 2.5.2.3.1.5

Multiply $- 2$ by $- 1$ .

$y = - x^{2} + 2 x + \frac{- 3}{- 1}$

Step 2.5.2.3.1.6

Divide $- 3$ by $- 1$ .

$y = - x^{2} + 2 x + 3$

Step 3

Replace $y$ with $f^{- 1} (x)$ to show the final answer.

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

Step 4

Verify if $f^{- 1} (x) = - x^{2} + 2 x + 3$ is the inverse of $f (x) = \sqrt{4 - x} + 1$ .

Tap for more steps...

Step 4.1

To verify the inverse, check if $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ .

Step 4.2

Evaluate $f^{- 1} (f (x))$ .

Tap for more steps...

Step 4.2.1

Set up the composite result function.

$f^{- 1} (f (x))$

Step 4.2.2

Evaluate $f^{- 1} (\sqrt{4 - x} + 1)$ by substituting in the value of $f$ into $f^{- 1}$ .

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

Step 4.2.3

Simplify each term.

Tap for more steps...

Step 4.2.3.1

Rewrite ${(\sqrt{4 - x} + 1)}^{2}$ as $(\sqrt{4 - x} + 1) (\sqrt{4 - x} + 1)$ .

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

Step 4.2.3.2

Expand $(\sqrt{4 - x} + 1) (\sqrt{4 - x} + 1)$ using the FOIL Method.

Tap for more steps...

Step 4.2.3.2.1

Apply the distributive property.

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

Step 4.2.3.2.2

Apply the distributive property.

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

Step 4.2.3.2.3

Apply the distributive property.

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

Step 4.2.3.3

Simplify and combine like terms.

Tap for more steps...

Step 4.2.3.3.1

Simplify each term.

Tap for more steps...

Step 4.2.3.3.1.1

Multiply $\sqrt{4 - x} \sqrt{4 - x}$ .

Tap for more steps...

Step 4.2.3.3.1.1.1

Raise $\sqrt{4 - x}$ to the power of $1$ .

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

Step 4.2.3.3.1.1.2

Raise $\sqrt{4 - x}$ to the power of $1$ .

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

Step 4.2.3.3.1.1.3

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

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

Step 4.2.3.3.1.1.4

Add $1$ and $1$ .

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

Step 4.2.3.3.1.2

Rewrite ${\sqrt{4 - x}}^{2}$ as $4 - x$ .

Tap for more steps...

Step 4.2.3.3.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 - x}$ as ${(4 - x)}^{\frac{1}{2}}$ .

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

Step 4.2.3.3.1.2.2

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

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

Step 4.2.3.3.1.2.3

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

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

Step 4.2.3.3.1.2.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.2.3.3.1.2.4.1

Cancel the common factor.

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

Step 4.2.3.3.1.2.4.2

Rewrite the expression.

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

Step 4.2.3.3.1.2.5

Simplify.

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

Step 4.2.3.3.1.3

Multiply $\sqrt{4 - x}$ by $1$ .

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

Step 4.2.3.3.1.4

Multiply $\sqrt{4 - x}$ by $1$ .

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

Step 4.2.3.3.1.5

Multiply $1$ by $1$ .

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

Step 4.2.3.3.2

Add $4$ and $1$ .

$f^{- 1} (\sqrt{4 - x} + 1) = - (5 - x + \sqrt{4 - x} + \sqrt{4 - x}) + 2 (\sqrt{4 - x} + 1) + 3$

Step 4.2.3.3.3

Add $\sqrt{4 - x}$ and $\sqrt{4 - x}$ .

$f^{- 1} (\sqrt{4 - x} + 1) = - (5 - x + 2 \sqrt{4 - x}) + 2 (\sqrt{4 - x} + 1) + 3$

Step 4.2.3.4

Apply the distributive property.

$f^{- 1} (\sqrt{4 - x} + 1) = - 1 \cdot 5 + x - (2 \sqrt{4 - x}) + 2 (\sqrt{4 - x} + 1) + 3$

Step 4.2.3.5

Simplify.

Tap for more steps...

Step 4.2.3.5.1

Multiply $- 1$ by $5$ .

$f^{- 1} (\sqrt{4 - x} + 1) = - 5 + x - (2 \sqrt{4 - x}) + 2 (\sqrt{4 - x} + 1) + 3$

Step 4.2.3.5.2

Multiply $- - x$ .

Tap for more steps...

Step 4.2.3.5.2.1

Multiply $- 1$ by $- 1$ .

$f^{- 1} (\sqrt{4 - x} + 1) = - 5 + 1 x - (2 \sqrt{4 - x}) + 2 (\sqrt{4 - x} + 1) + 3$

Step 4.2.3.5.2.2

Multiply $x$ by $1$ .

$f^{- 1} (\sqrt{4 - x} + 1) = - 5 + x - (2 \sqrt{4 - x}) + 2 (\sqrt{4 - x} + 1) + 3$

Step 4.2.3.5.3

Multiply $2$ by $- 1$ .

$f^{- 1} (\sqrt{4 - x} + 1) = - 5 + x - 2 \sqrt{4 - x} + 2 (\sqrt{4 - x} + 1) + 3$

Step 4.2.3.6

Apply the distributive property.

$f^{- 1} (\sqrt{4 - x} + 1) = - 5 + x - 2 \sqrt{4 - x} + 2 \sqrt{4 - x} + 2 \cdot 1 + 3$

Step 4.2.3.7

Multiply $2$ by $1$ .

$f^{- 1} (\sqrt{4 - x} + 1) = - 5 + x - 2 \sqrt{4 - x} + 2 \sqrt{4 - x} + 2 + 3$

Step 4.2.4

Simplify by adding terms.

Tap for more steps...

Step 4.2.4.1

Combine the opposite terms in $- 5 + x - 2 \sqrt{4 - x} + 2 \sqrt{4 - x} + 2 + 3$ .

Tap for more steps...

Step 4.2.4.1.1

Add $- 2 \sqrt{4 - x}$ and $2 \sqrt{4 - x}$ .

$f^{- 1} (\sqrt{4 - x} + 1) = - 5 + x + 0 + 2 + 3$

Step 4.2.4.1.2

Add $- 5 + x$ and $0$ .

$f^{- 1} (\sqrt{4 - x} + 1) = - 5 + x + 2 + 3$

Step 4.2.4.2

Add $- 5$ and $2$ .

$f^{- 1} (\sqrt{4 - x} + 1) = x - 3 + 3$

Step 4.2.4.3

Combine the opposite terms in $x - 3 + 3$ .

Tap for more steps...

Step 4.2.4.3.1

Add $- 3$ and $3$ .

$f^{- 1} (\sqrt{4 - x} + 1) = x + 0$

Step 4.2.4.3.2

Add $x$ and $0$ .

$f^{- 1} (\sqrt{4 - x} + 1) = x$

Step 4.3

Evaluate $f (f^{- 1} (x))$ .

Tap for more steps...

Step 4.3.1

Set up the composite result function.

$f (f^{- 1} (x))$

Step 4.3.2

Evaluate $f (- x^{2} + 2 x + 3)$ by substituting in the value of $f^{- 1}$ into $f$ .

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

Step 4.3.3

Simplify each term.

Tap for more steps...

Step 4.3.3.1

Apply the distributive property.

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

Step 4.3.3.2

Simplify.

Tap for more steps...

Step 4.3.3.2.1

Multiply $- - x^{2}$ .

Tap for more steps...

Step 4.3.3.2.1.1

Multiply $- 1$ by $- 1$ .

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

Step 4.3.3.2.1.2

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

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

Step 4.3.3.2.2

Multiply $2$ by $- 1$ .

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

Step 4.3.3.2.3

Multiply $- 1$ by $3$ .

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

Step 4.3.3.3

Subtract $3$ from $4$ .

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

Step 4.3.3.4

Factor using the perfect square rule.

Tap for more steps...

Step 4.3.3.4.1

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

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

Step 4.3.3.4.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 x = 2 \cdot x \cdot 1$

Step 4.3.3.4.3

Rewrite the polynomial.

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

Step 4.3.3.4.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 1$ .

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

Step 4.3.3.5

Pull terms out from under the radical, assuming positive real numbers.

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

Step 4.3.4

Combine the opposite terms in $x - 1 + 1$ .

Tap for more steps...

Step 4.3.4.1

Add $- 1$ and $1$ .

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

Step 4.3.4.2

Add $x$ and $0$ .

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

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - x^{2} + 2 x + 3$ is the inverse of $f (x) = \sqrt{4 - x} + 1$ .

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