Algebra Examples

$y = \arcsin (2 x)$

Step 1

Set the argument in $\arcsin (2 x)$ greater than or equal to $- 1$ to find where the expression is defined.

$2 x \geq - 1$

Step 2

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

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Step 2.1

Divide each term in $2 x \geq - 1$ by $2$ .

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $2$ .

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Step 2.2.1.1

Cancel the common factor.

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

Step 2.2.1.2

Divide $x$ by $1$ .

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

Step 2.3

Simplify the right side.

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Step 2.3.1

Move the negative in front of the fraction.

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

Step 3

Set the argument in $\arcsin (2 x)$ less than or equal to $1$ to find where the expression is defined.

$2 x \leq 1$

Step 4

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

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Step 4.1

Divide each term in $2 x \leq 1$ by $2$ .

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

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $2$ .

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Step 4.2.1.1

Cancel the common factor.

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

Step 4.2.1.2

Divide $x$ by $1$ .

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

Step 5

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

Interval Notation:

$[- \frac{1}{2}, \frac{1}{2}]$

Set-Builder Notation:

${x | - \frac{1}{2} \leq x \leq \frac{1}{2}}$

Step 6

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

Interval Notation:

$[- \frac{π}{2}, \frac{π}{2}]$

Set-Builder Notation:

${y | - \frac{π}{2} \leq y \leq \frac{π}{2}}$

Step 7

Determine the domain and range.

Domain: $[- \frac{1}{2}, \frac{1}{2}], {x | - \frac{1}{2} \leq x \leq \frac{1}{2}}$

Range: $[- \frac{π}{2}, \frac{π}{2}], {y | - \frac{π}{2} \leq y \leq \frac{π}{2}}$

Step 8