Trigonometry Examples

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

Step 1

Interchange the variables.

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

Step 2

Solve for $y$ .

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Step 2.1

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

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

Step 2.2

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

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

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

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

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

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

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Step 2.3.2.1.1

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

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Step 2.3.2.1.1.1

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

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

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.

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

Step 2.3.2.1.1.2.2

Rewrite the expression.

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

Step 2.3.2.1.2

Simplify.

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

Step 2.4

Solve for $y$ .

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Step 2.4.1

Subtract $4$ from both sides of the equation.

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

Step 2.4.2

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

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Step 2.4.2.1

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

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

Step 2.4.2.2

Simplify the left side.

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Step 2.4.2.2.1

Cancel the common factor of $7$ .

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Step 2.4.2.2.1.1

Cancel the common factor.

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

Step 2.4.2.2.1.2

Divide $y$ by $1$ .

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

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.

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

Step 3

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

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

Step 4

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

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

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

Step 4.2.3

Combine the numerators over the common denominator.

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

Step 4.2.4

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

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Step 4.2.4.1

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

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

Step 4.2.4.2

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

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

Step 4.2.4.3

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

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

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.

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

Step 4.2.4.4.2

Rewrite the expression.

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

Step 4.2.4.5

Simplify.

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

Step 4.2.5

Simplify terms.

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Step 4.2.5.1

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

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Step 4.2.5.1.1

Subtract $4$ from $4$ .

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

Step 4.2.5.1.2

Add $7 x$ and $0$ .

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

Step 4.2.5.2

Cancel the common factor of $7$ .

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Step 4.2.5.2.1

Cancel the common factor.

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

Step 4.2.5.2.2

Divide $x$ by $1$ .

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

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

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

Step 4.3.3

Apply the distributive property.

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

Step 4.3.4

Cancel the common factor of $7$ .

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Step 4.3.4.1

Cancel the common factor.

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

Step 4.3.4.2

Rewrite the expression.

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

Step 4.3.5

Cancel the common factor of $7$ .

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Step 4.3.5.1

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

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

Step 4.3.5.2

Cancel the common factor.

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

Step 4.3.5.3

Rewrite the expression.

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

Step 4.3.6

Simplify by adding numbers.

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Step 4.3.6.1

Add $- 4$ and $4$ .

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

Step 4.3.6.2

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

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

Step 4.3.7

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

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

Step 4.4

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

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