Precalculus Examples

$f (x) = \frac{1 + \sqrt{1 - x}}{\sqrt{4 x - 2 x^{2}}}$

Step 1

Set the radicand in $\sqrt{1 - x}$ greater than or equal to $0$ to find where the expression is defined.

$1 - x \geq 0$

Step 2

Solve for $x$ .

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Step 2.1

Subtract $1$ from both sides of the inequality.

$- x \geq - 1$

Step 2.2

Divide each term in $- x \geq - 1$ by $- 1$ and simplify.

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Step 2.2.1

Divide each term in $- x \geq - 1$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \leq \frac{- 1}{- 1}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \leq \frac{- 1}{- 1}$

Step 2.2.2.2

Divide $x$ by $1$ .

$x \leq \frac{- 1}{- 1}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Divide $- 1$ by $- 1$ .

$x \leq 1$

Step 3

Set the radicand in $\sqrt{4 x - 2 x^{2}}$ greater than or equal to $0$ to find where the expression is defined.

$4 x - 2 x^{2} \geq 0$

Step 4

Solve for $x$ .

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Step 4.1

Convert the inequality to an equation.

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

Step 4.2

Factor $2 x$ out of $4 x - 2 x^{2}$ .

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Step 4.2.1

Factor $2 x$ out of $4 x$ .

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

Step 4.2.2

Factor $2 x$ out of $- 2 x^{2}$ .

$2 x (2) + 2 x (- x) = 0$

Step 4.2.3

Factor $2 x$ out of $2 x (2) + 2 x (- x)$ .

$2 x (2 - x) = 0$

Step 4.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$2 - x = 0$

Step 4.4

Set $x$ equal to $0$ .

$x = 0$

Step 4.5

Set $2 - x$ equal to $0$ and solve for $x$ .

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Step 4.5.1

Set $2 - x$ equal to $0$ .

$2 - x = 0$

Step 4.5.2

Solve $2 - x = 0$ for $x$ .

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Step 4.5.2.1

Subtract $2$ from both sides of the equation.

$- x = - 2$

Step 4.5.2.2

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

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Step 4.5.2.2.1

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

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

Step 4.5.2.2.2

Simplify the left side.

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Step 4.5.2.2.2.1

Dividing two negative values results in a positive value.

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

Step 4.5.2.2.2.2

Divide $x$ by $1$ .

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

Step 4.5.2.2.3

Simplify the right side.

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Step 4.5.2.2.3.1

Divide $- 2$ by $- 1$ .

$x = 2$

Step 4.6

The final solution is all the values that make $2 x (2 - x) = 0$ true.

$x = 0, 2$

Step 4.7

Use each root to create test intervals.

$x < 0$

$0 < x < 2$

$x > 2$

Step 4.8

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 4.8.1

Test a value on the interval $x < 0$ to see if it makes the inequality true.

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Step 4.8.1.1

Choose a value on the interval $x < 0$ and see if this value makes the original inequality true.

$x = - 2$

Step 4.8.1.2

Replace $x$ with $- 2$ in the original inequality.

$4 (- 2) - 2 {(- 2)}^{2} \geq 0$

Step 4.8.1.3

The left side $- 16$ is less than the right side $0$ , which means that the given statement is false.

False

Step 4.8.2

Test a value on the interval $0 < x < 2$ to see if it makes the inequality true.

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Step 4.8.2.1

Choose a value on the interval $0 < x < 2$ and see if this value makes the original inequality true.

$x = 1$

Step 4.8.2.2

Replace $x$ with $1$ in the original inequality.

$4 (1) - 2 {(1)}^{2} \geq 0$

Step 4.8.2.3

The left side $2$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 4.8.3

Test a value on the interval $x > 2$ to see if it makes the inequality true.

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Step 4.8.3.1

Choose a value on the interval $x > 2$ and see if this value makes the original inequality true.

$x = 4$

Step 4.8.3.2

Replace $x$ with $4$ in the original inequality.

$4 (4) - 2 {(4)}^{2} \geq 0$

Step 4.8.3.3

The left side $- 16$ is less than the right side $0$ , which means that the given statement is false.

False

Step 4.8.4

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

$x < 0$ False

$0 < x < 2$ True

$x > 2$ False

$x < 0$ False

$0 < x < 2$ True

$x > 2$ False

Step 4.9

The solution consists of all of the true intervals.

$0 \leq x \leq 2$

Step 5

Set the denominator in $\frac{1 + \sqrt{1 - x}}{\sqrt{4 x - 2 x^{2}}}$ equal to $0$ to find where the expression is undefined.

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

Step 6

Solve for $x$ .

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Step 6.1

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

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

Step 6.2

Simplify each side of the equation.

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Step 6.2.1

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

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

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

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

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Step 6.2.2.1.1

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

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Step 6.2.2.1.1.1

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

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

Step 6.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 6.2.2.1.1.2.1

Cancel the common factor.

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

Step 6.2.2.1.1.2.2

Rewrite the expression.

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

Step 6.2.2.1.2

Simplify.

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

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

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

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

Step 6.3

Solve for $x$ .

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Step 6.3.1

Factor the left side of the equation.

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Step 6.3.1.1

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

$4 u - 2 u^{2} = 0$

Step 6.3.1.2

Factor $2 u$ out of $4 u - 2 u^{2}$ .

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Step 6.3.1.2.1

Factor $2 u$ out of $4 u$ .

$2 u (2) - 2 u^{2} = 0$

Step 6.3.1.2.2

Factor $2 u$ out of $- 2 u^{2}$ .

$2 u (2) + 2 u (- u) = 0$

Step 6.3.1.2.3

Factor $2 u$ out of $2 u (2) + 2 u (- u)$ .

$2 u (2 - u) = 0$

Step 6.3.1.3

Replace all occurrences of $u$ with $x$ .

$2 x (2 - x) = 0$

Step 6.3.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$2 - x = 0$

Step 6.3.3

Set $x$ equal to $0$ .

$x = 0$

Step 6.3.4

Set $2 - x$ equal to $0$ and solve for $x$ .

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Step 6.3.4.1

Set $2 - x$ equal to $0$ .

$2 - x = 0$

Step 6.3.4.2

Solve $2 - x = 0$ for $x$ .

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Step 6.3.4.2.1

Subtract $2$ from both sides of the equation.

$- x = - 2$

Step 6.3.4.2.2

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

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Step 6.3.4.2.2.1

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

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

Step 6.3.4.2.2.2

Simplify the left side.

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Step 6.3.4.2.2.2.1

Dividing two negative values results in a positive value.

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

Step 6.3.4.2.2.2.2

Divide $x$ by $1$ .

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

Step 6.3.4.2.2.3

Simplify the right side.

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Step 6.3.4.2.2.3.1

Divide $- 2$ by $- 1$ .

$x = 2$

Step 6.3.5

The final solution is all the values that make $2 x (2 - x) = 0$ true.

$x = 0, 2$

Step 7

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(0, 1]$

Set-Builder Notation:

${x | 0 < x \leq 1}$

Step 8