Precalculus Examples

$y = 3 - \sqrt{2 - x}$

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 $x$ .

$0 = 3 - \sqrt{2 - x}$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $3 - \sqrt{2 - x} = 0$ .

$3 - \sqrt{2 - x} = 0$

Step 1.2.2

Subtract $3$ from both sides of the equation.

$- \sqrt{2 - x} = - 3$

Step 1.2.3

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

${(- \sqrt{2 - x})}^{2} = {(- 3)}^{2}$

Step 1.2.4

Simplify each side of the equation.

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Step 1.2.4.1

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

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

Step 1.2.4.2

Simplify the left side.

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Step 1.2.4.2.1

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

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Step 1.2.4.2.1.1

Apply the product rule to $- {(2 - x)}^{\frac{1}{2}}$ .

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

Step 1.2.4.2.1.2

Raise $- 1$ to the power of $2$ .

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

Step 1.2.4.2.1.3

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

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

Step 1.2.4.2.1.4

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

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Step 1.2.4.2.1.4.1

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

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

Step 1.2.4.2.1.4.2

Cancel the common factor of $2$ .

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Step 1.2.4.2.1.4.2.1

Cancel the common factor.

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

Step 1.2.4.2.1.4.2.2

Rewrite the expression.

${(2 - x)}^{1} = {(- 3)}^{2}$

Step 1.2.4.2.1.5

Simplify.

$2 - x = {(- 3)}^{2}$

Step 1.2.4.3

Simplify the right side.

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Step 1.2.4.3.1

Raise $- 3$ to the power of $2$ .

$2 - x = 9$

Step 1.2.5

Solve for $x$ .

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Step 1.2.5.1

Move all terms not containing $x$ to the right side of the equation.

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Step 1.2.5.1.1

Subtract $2$ from both sides of the equation.

$- x = 9 - 2$

Step 1.2.5.1.2

Subtract $2$ from $9$ .

$- x = 7$

Step 1.2.5.2

Divide each term in $- x = 7$ by $- 1$ and simplify.

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Step 1.2.5.2.1

Divide each term in $- x = 7$ by $- 1$ .

$\frac{- x}{- 1} = \frac{7}{- 1}$

Step 1.2.5.2.2

Simplify the left side.

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Step 1.2.5.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{7}{- 1}$

Step 1.2.5.2.2.2

Divide $x$ by $1$ .

$x = \frac{7}{- 1}$

Step 1.2.5.2.3

Simplify the right side.

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Step 1.2.5.2.3.1

Divide $7$ by $- 1$ .

$x = - 7$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(- 7, 0)$

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 $y$ .

$y = 3 - \sqrt{2 - (0)}$

Step 2.2

Simplify each term.

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Step 2.2.1

Multiply $- 1$ by $0$ .

$y = 3 - \sqrt{2 + 0}$

Step 2.2.2

Add $2$ and $0$ .

$y = 3 - \sqrt{2}$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 3 - \sqrt{2})$

Step 3

List the intersections.

x-intercept(s): $(- 7, 0)$

y-intercept(s): $(0, 3 - \sqrt{2})$

Step 4