Precalculus Examples

$f (x) = \sqrt{4 - x}$

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 $x$ .

$0 = \sqrt{4 - x}$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $\sqrt{4 - x} = 0$ .

$\sqrt{4 - x} = 0$

Step 1.2.2

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

${\sqrt{4 - x}}^{2} = 0^{2}$

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

${({(4 - x)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Simplify ${({(4 - x)}^{\frac{1}{2}})}^{2}$ .

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Step 1.2.3.2.1.1

Multiply the exponents in ${({(4 - x)}^{\frac{1}{2}})}^{2}$ .

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Step 1.2.3.2.1.1.1

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

${(4 - x)}^{\frac{1}{2} \cdot 2} = 0^{2}$

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.

${(4 - x)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 1.2.3.2.1.1.2.2

Rewrite the expression.

${(4 - x)}^{1} = 0^{2}$

Step 1.2.3.2.1.2

Simplify.

$4 - x = 0^{2}$

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 $0$ .

$4 - x = 0$

Step 1.2.4

Solve for $x$ .

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Step 1.2.4.1

Subtract $4$ from both sides of the equation.

$- x = - 4$

Step 1.2.4.2

Divide each term in $- x = - 4$ by $- 1$ and simplify.

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Step 1.2.4.2.1

Divide each term in $- x = - 4$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 4}{- 1}$

Step 1.2.4.2.2

Simplify the left side.

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Step 1.2.4.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 4}{- 1}$

Step 1.2.4.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 4}{- 1}$

Step 1.2.4.2.3

Simplify the right side.

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Step 1.2.4.2.3.1

Divide $- 4$ by $- 1$ .

$x = 4$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(4, 0)$

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 $y$ .

$y = \sqrt{4 - (0)}$

Step 2.2

Simplify $\sqrt{4 - (0)}$ .

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Step 2.2.1

Multiply $- 1$ by $0$ .

$y = \sqrt{4 + 0}$

Step 2.2.2

Add $4$ and $0$ .

$y = \sqrt{4}$

Step 2.2.3

Rewrite $4$ as $2^{2}$ .

$y = \sqrt{2^{2}}$

Step 2.2.4

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

$y = 2$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 2)$

Step 3

List the intersections.

x-intercept(s): $(4, 0)$

y-intercept(s): $(0, 2)$

Step 4