Find the Domain and Range y = square root of 2x-x^2

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

Step 1

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

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

Step 2

Solve for $x$ .

Convert the inequality to an equation.

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

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

Factor $x$ out of $2 x$ .

$x \cdot 2 - x^{2} = 0$

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

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

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

$x (2 - x) = 0$

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

$x = 0$

$2 - x = 0$

Set $x$ equal to $0$ .

$x = 0$

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

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

$2 - x = 0$

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

Subtract $2$ from both sides of the equation.

$- x = - 2$

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

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

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

Simplify the left side.

Dividing two negative values results in a positive value.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Divide $- 2$ by $- 1$ .

$x = 2$

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

$x = 0, 2$

Use each root to create test intervals.

$x < 0$

$0 < x < 2$

$x > 2$

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

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

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

$x = - 2$

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

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

The left side $- 8$ is less than the right side $0$ , which means that the given statement is false.

False

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

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

$x = 1$

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

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

The left side $1$ is greater than the right side $0$ , which means that the given statement is always true.

True

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

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

$x = 4$

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

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

The left side $- 8$ is less than the right side $0$ , which means that the given statement is false.

False

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

The solution consists of all of the true intervals.

$0 \leq x \leq 2$

Step 3

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

Interval Notation:

$[0, 2]$

Set-Builder Notation:

${x | 0 \leq x \leq 2}$

Step 4

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$[0, 1]$

Set-Builder Notation:

${y | 0 \leq y \leq 1}$

Step 5

Determine the domain and range.

Domain: $[0, 2], {x | 0 \leq x \leq 2}$

Range: $[0, 1], {y | 0 \leq y \leq 1}$

Step 6

Find the Domain and Range y = square root of 2x-x^2

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