Trigonometry Examples

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

Step 1

Write $f (x) = \sqrt{4 x^{2} + 3}$ as an equation.

$y = \sqrt{4 x^{2} + 3}$

Step 2

Interchange the variables.

$x = \sqrt{4 y^{2} + 3}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\sqrt{4 y^{2} + 3} = x$ .

$\sqrt{4 y^{2} + 3} = x$

Step 3.2

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

${\sqrt{4 y^{2} + 3}}^{2} = x^{2}$

Step 3.3

Simplify each side of the equation.

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Step 3.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 y^{2} + 3}$ as ${(4 y^{2} + 3)}^{\frac{1}{2}}$ .

${({(4 y^{2} + 3)}^{\frac{1}{2}})}^{2} = x^{2}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Simplify ${({(4 y^{2} + 3)}^{\frac{1}{2}})}^{2}$ .

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Step 3.3.2.1.1

Multiply the exponents in ${({(4 y^{2} + 3)}^{\frac{1}{2}})}^{2}$ .

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Step 3.3.2.1.1.1

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

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

Step 3.3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.3.2.1.1.2.1

Cancel the common factor.

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

Step 3.3.2.1.1.2.2

Rewrite the expression.

${(4 y^{2} + 3)}^{1} = x^{2}$

Step 3.3.2.1.2

Simplify.

$4 y^{2} + 3 = x^{2}$

Step 3.4

Solve for $y$ .

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Step 3.4.1

Subtract $3$ from both sides of the equation.

$4 y^{2} = x^{2} - 3$

Step 3.4.2

Divide each term in $4 y^{2} = x^{2} - 3$ by $4$ and simplify.

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Step 3.4.2.1

Divide each term in $4 y^{2} = x^{2} - 3$ by $4$ .

$\frac{4 y^{2}}{4} = \frac{x^{2}}{4} + \frac{- 3}{4}$

Step 3.4.2.2

Simplify the left side.

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Step 3.4.2.2.1

Cancel the common factor of $4$ .

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Step 3.4.2.2.1.1

Cancel the common factor.

$\frac{4 y^{2}}{4} = \frac{x^{2}}{4} + \frac{- 3}{4}$

Step 3.4.2.2.1.2

Divide $y^{2}$ by $1$ .

$y^{2} = \frac{x^{2}}{4} + \frac{- 3}{4}$

Step 3.4.2.3

Simplify the right side.

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Step 3.4.2.3.1

Move the negative in front of the fraction.

$y^{2} = \frac{x^{2}}{4} - \frac{3}{4}$

Step 3.4.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \pm \sqrt{\frac{x^{2}}{4} - \frac{3}{4}}$

Step 3.4.4

Simplify $\pm \sqrt{\frac{x^{2}}{4} - \frac{3}{4}}$ .

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Step 3.4.4.1

Combine the numerators over the common denominator.

$y = \pm \sqrt{\frac{x^{2} - 3}{4}}$

Step 3.4.4.2

Rewrite $\frac{x^{2} - 3}{4}$ as ${(\frac{1}{2})}^{2} (x^{2} - 3)$ .

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Step 3.4.4.2.1

Factor the perfect power $1^{2}$ out of $x^{2} - 3$ .

$y = \pm \sqrt{\frac{1^{2} (x^{2} - 3)}{4}}$

Step 3.4.4.2.2

Factor the perfect power $2^{2}$ out of $4$ .

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

Step 3.4.4.2.3

Rearrange the fraction $\frac{1^{2} (x^{2} - 3)}{2^{2} \cdot 1}$ .

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

Step 3.4.4.3

Pull terms out from under the radical.

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

Step 3.4.4.4

Combine $\frac{1}{2}$ and $\sqrt{x^{2} - 3}$ .

$y = \pm \frac{\sqrt{x^{2} - 3}}{2}$

Step 3.4.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.4.5.1

First, use the positive value of the $\pm$ to find the first solution.