Calculus Examples

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

Step 1

Set the radicand in $\sqrt{y - x^{2}}$ greater than or equal to $0$ to find where the expression is defined.

$y - x^{2} \geq 0$

Step 2

Solve for $y$ .

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Step 2.1

Add $x^{2}$ to both sides of the inequality.

$y \geq x^{2}$

Step 2.2

Use each root to create test intervals.

$y < x^{2}$

$y > x^{2}$

Step 2.3

Compare the intervals to determine which ones satisfy the original inequality.

Step 2.4

Since there are no numbers that fall within the interval, this inequality has no solution.

No solution

Step 3

Set the denominator in $\frac{1}{\sqrt{y - x^{2}}}$ equal to $0$ to find where the expression is undefined.

$\sqrt{y - x^{2}} = 0$

Step 4

Solve for $y$ .

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Step 4.1

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

${\sqrt{y - x^{2}}}^{2} = 0^{2}$

Step 4.2

Simplify each side of the equation.

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Step 4.2.1

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

${({(y - x^{2})}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

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

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Step 4.2.2.1.1

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

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Step 4.2.2.1.1.1

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

${(y - x^{2})}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 4.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 4.2.2.1.1.2.1

Cancel the common factor.

${(y - x^{2})}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 4.2.2.1.1.2.2

Rewrite the expression.

${(y - x^{2})}^{1} = 0^{2}$

Step 4.2.2.1.2

Simplify.

$y - x^{2} = 0^{2}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Raising $0$ to any positive power yields $0$ .

$y - x^{2} = 0$

Step 4.3

Add $x^{2}$ to both sides of the equation.

$y = x^{2}$

Step 5

The domain is all real numbers.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${y | y \in ℝ}$

Step 6