Trigonometry Examples

$\sqrt{7 + 4 x}$

Step 1

Interchange the variables.

$x = \sqrt{7 + 4 y}$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $\sqrt{7 + 4 y} = x$ .

$\sqrt{7 + 4 y} = x$

Step 2.2

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

${\sqrt{7 + 4 y}}^{2} = x^{2}$

Step 2.3

Simplify each side of the equation.

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Step 2.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7 + 4 y}$ as ${(7 + 4 y)}^{\frac{1}{2}}$ .

${({(7 + 4 y)}^{\frac{1}{2}})}^{2} = x^{2}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Simplify ${({(7 + 4 y)}^{\frac{1}{2}})}^{2}$ .

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Step 2.3.2.1.1

Multiply the exponents in ${({(7 + 4 y)}^{\frac{1}{2}})}^{2}$ .

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Step 2.3.2.1.1.1

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

${(7 + 4 y)}^{\frac{1}{2} \cdot 2} = x^{2}$

Step 2.3.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.3.2.1.1.2.1

Cancel the common factor.

${(7 + 4 y)}^{\frac{1}{2} \cdot 2} = x^{2}$

Step 2.3.2.1.1.2.2

Rewrite the expression.

${(7 + 4 y)}^{1} = x^{2}$

Step 2.3.2.1.2

Simplify.

$7 + 4 y = x^{2}$

Step 2.4

Solve for $y$ .

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Step 2.4.1

Subtract $7$ from both sides of the equation.

$4 y = x^{2} - 7$

Step 2.4.2

Divide each term in $4 y = x^{2} - 7$ by $4$ and simplify.

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Step 2.4.2.1

Divide each term in $4 y = x^{2} - 7$ by $4$ .

$\frac{4 y}{4} = \frac{x^{2}}{4} + \frac{- 7}{4}$

Step 2.4.2.2

Simplify the left side.

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Step 2.4.2.2.1

Cancel the common factor of $4$ .

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Step 2.4.2.2.1.1

Cancel the common factor.

$\frac{4 y}{4} = \frac{x^{2}}{4} + \frac{- 7}{4}$

Step 2.4.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{x^{2}}{4} + \frac{- 7}{4}$

Step 2.4.2.3

Simplify the right side.

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Step 2.4.2.3.1

Move the negative in front of the fraction.

$y = \frac{x^{2}}{4} - \frac{7}{4}$

Step 3

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

$f^{- 1} (x) = \frac{x^{2}}{4} - \frac{7}{4}$

Step 4

Verify if $f^{- 1} (x) = \frac{x^{2}}{4} - \frac{7}{4}$ is the inverse of $f (x) = \sqrt{7 + 4 x}$ .

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Step 4.1

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

Step 4.2

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

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Step 4.2.1

Set up the composite result function.

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

Step 4.2.2

Evaluate $f^{- 1} (\sqrt{7 + 4 x})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\sqrt{7 + 4 x}) = \frac{{(\sqrt{7 + 4 x})}^{2}}{4} - \frac{7}{4}$

Step 4.2.3

Combine the numerators over the common denominator.

$f^{- 1} (\sqrt{7 + 4 x}) = \frac{{(\sqrt{7 + 4 x})}^{2} - 7}{4}$

Step 4.2.4

Rewrite ${\sqrt{7 + 4 x}}^{2}$ as $7 + 4 x$ .

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Step 4.2.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7 + 4 x}$ as ${(7 + 4 x)}^{\frac{1}{2}}$ .

$f^{- 1} (\sqrt{7 + 4 x}) = \frac{{({(7 + 4 x)}^{\frac{1}{2}})}^{2} - 7}{4}$

Step 4.2.4.2

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

$f^{- 1} (\sqrt{7 + 4 x}) = \frac{{(7 + 4 x)}^{\frac{1}{2} \cdot 2} - 7}{4}$

Step 4.2.4.3

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

$f^{- 1} (\sqrt{7 + 4 x}) = \frac{{(7 + 4 x)}^{\frac{2}{2}} - 7}{4}$

Step 4.2.4.4

Cancel the common factor of $2$ .

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Step 4.2.4.4.1

Cancel the common factor.

$f^{- 1} (\sqrt{7 + 4 x}) = \frac{{(7 + 4 x)}^{\frac{2}{2}} - 7}{4}$

Step 4.2.4.4.2

Rewrite the expression.

$f^{- 1} (\sqrt{7 + 4 x}) = \frac{(7 + 4 x) - 7}{4}$

Step 4.2.4.5

Simplify.

$f^{- 1} (\sqrt{7 + 4 x}) = \frac{7 + 4 x - 7}{4}$

Step 4.2.5

Simplify terms.

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Step 4.2.5.1

Combine the opposite terms in $7 + 4 x - 7$ .

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Step 4.2.5.1.1

Subtract $7$ from $7$ .

$f^{- 1} (\sqrt{7 + 4 x}) = \frac{4 x + 0}{4}$

Step 4.2.5.1.2

Add $4 x$ and $0$ .

$f^{- 1} (\sqrt{7 + 4 x}) = \frac{4 x}{4}$

Step 4.2.5.2

Cancel the common factor of $4$ .

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Step 4.2.5.2.1

Cancel the common factor.

$f^{- 1} (\sqrt{7 + 4 x}) = \frac{4 x}{4}$

Step 4.2.5.2.2

Divide $x$ by $1$ .

$f^{- 1} (\sqrt{7 + 4 x}) = x$

Step 4.3

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

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Step 4.3.1

Set up the composite result function.

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

Step 4.3.2

Evaluate $f (\frac{x^{2}}{4} - \frac{7}{4})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{x^{2}}{4} - \frac{7}{4}) = \sqrt{7 + 4 (\frac{x^{2}}{4} - \frac{7}{4})}$

Step 4.3.3

Apply the distributive property.

$f (\frac{x^{2}}{4} - \frac{7}{4}) = \sqrt{7 + 4 (\frac{x^{2}}{4}) + 4 (- \frac{7}{4})}$

Step 4.3.4

Cancel the common factor of $4$ .

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Step 4.3.4.1

Cancel the common factor.

$f (\frac{x^{2}}{4} - \frac{7}{4}) = \sqrt{7 + 4 (\frac{x^{2}}{4}) + 4 (- \frac{7}{4})}$

Step 4.3.4.2

Rewrite the expression.

$f (\frac{x^{2}}{4} - \frac{7}{4}) = \sqrt{7 + x^{2} + 4 (- \frac{7}{4})}$

Step 4.3.5

Cancel the common factor of $4$ .

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Step 4.3.5.1

Move the leading negative in $- \frac{7}{4}$ into the numerator.

$f (\frac{x^{2}}{4} - \frac{7}{4}) = \sqrt{7 + x^{2} + 4 (\frac{- 7}{4})}$

Step 4.3.5.2

Cancel the common factor.

$f (\frac{x^{2}}{4} - \frac{7}{4}) = \sqrt{7 + x^{2} + 4 (\frac{- 7}{4})}$

Step 4.3.5.3

Rewrite the expression.

$f (\frac{x^{2}}{4} - \frac{7}{4}) = \sqrt{7 + x^{2} - 7}$

Step 4.3.6

Simplify by subtracting numbers.

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Step 4.3.6.1

Subtract $7$ from $7$ .

$f (\frac{x^{2}}{4} - \frac{7}{4}) = \sqrt{x^{2} + 0}$

Step 4.3.6.2

Add $x^{2}$ and $0$ .

$f (\frac{x^{2}}{4} - \frac{7}{4}) = \sqrt{x^{2}}$

Step 4.3.7

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

$f (\frac{x^{2}}{4} - \frac{7}{4}) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{x^{2}}{4} - \frac{7}{4}$ is the inverse of $f (x) = \sqrt{7 + 4 x}$ .

$f^{- 1} (x) = \frac{x^{2}}{4} - \frac{7}{4}$