Precalculus Examples

$y = \sqrt{2 x - 1}$

Step 1

Find the x-intercepts.

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Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = \sqrt{2 x - 1}$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $\sqrt{2 x - 1} = 0$ .

$\sqrt{2 x - 1} = 0$

Step 1.2.2

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

${\sqrt{2 x - 1}}^{2} = 0^{2}$

Step 1.2.3

Simplify each side of the equation.

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Step 1.2.3.1

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

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

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

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

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Step 1.2.3.2.1.1

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

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Step 1.2.3.2.1.1.1

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

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

Step 1.2.3.2.1.1.2

Cancel the common factor of $2$ .

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Step 1.2.3.2.1.1.2.1

Cancel the common factor.

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

Step 1.2.3.2.1.1.2.2

Rewrite the expression.

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

Step 1.2.3.2.1.2

Simplify.

$2 x - 1 = 0^{2}$

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

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

$2 x - 1 = 0$

Step 1.2.4

Solve for $x$ .

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Step 1.2.4.1

Add $1$ to both sides of the equation.

$2 x = 1$

Step 1.2.4.2

Divide each term in $2 x = 1$ by $2$ and simplify.

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Step 1.2.4.2.1

Divide each term in $2 x = 1$ by $2$ .

$\frac{2 x}{2} = \frac{1}{2}$

Step 1.2.4.2.2

Simplify the left side.

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Step 1.2.4.2.2.1

Cancel the common factor of $2$ .

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Step 1.2.4.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{1}{2}$

Step 1.2.4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{2}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\frac{1}{2}, 0)$

Step 2

Find the y-intercepts.

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Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = \sqrt{2 (0) - 1}$

Step 2.2

Simplify $\sqrt{2 (0) - 1}$ .

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Step 2.2.1

Multiply $2$ by $0$ .

$y = \sqrt{0 - 1}$

Step 2.2.2

Subtract $1$ from $0$ .

$y = \sqrt{- 1}$

Step 2.2.3

Rewrite $\sqrt{- 1}$ as $i$ .

$y = i$

Step 2.3

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

y-intercept(s): $None$

Step 3

List the intersections.

x-intercept(s): $(\frac{1}{2}, 0)$

y-intercept(s): $None$

Step 4