Calculus Examples

$f (x) = \arcsin (4 - 2 x)$

Step 1

Set the argument in $\arcsin (4 - 2 x)$ greater than or equal to $- 1$ to find where the expression is defined.

$4 - 2 x \geq - 1$

Step 2

Solve for $x$ .

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Step 2.1

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

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Step 2.1.1

Subtract $4$ from both sides of the inequality.

$- 2 x \geq - 1 - 4$

Step 2.1.2

Subtract $4$ from $- 1$ .

$- 2 x \geq - 5$

Step 2.2

Divide each term in $- 2 x \geq - 5$ by $- 2$ and simplify.

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Step 2.2.1

Divide each term in $- 2 x \geq - 5$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 x}{- 2} \leq \frac{- 5}{- 2}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $- 2$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{- 2 x}{- 2} \leq \frac{- 5}{- 2}$

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x \leq \frac{- 5}{- 2}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Dividing two negative values results in a positive value.

$x \leq \frac{5}{2}$

Step 3

Set the argument in $\arcsin (4 - 2 x)$ less than or equal to $1$ to find where the expression is defined.

$4 - 2 x \leq 1$

Step 4

Solve for $x$ .

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Step 4.1

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

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Step 4.1.1

Subtract $4$ from both sides of the inequality.

$- 2 x \leq 1 - 4$

Step 4.1.2

Subtract $4$ from $1$ .

$- 2 x \leq - 3$

Step 4.2

Divide each term in $- 2 x \leq - 3$ by $- 2$ and simplify.

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Step 4.2.1

Divide each term in $- 2 x \leq - 3$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 x}{- 2} \geq \frac{- 3}{- 2}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $- 2$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{- 2 x}{- 2} \geq \frac{- 3}{- 2}$

Step 4.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{- 3}{- 2}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Dividing two negative values results in a positive value.

$x \geq \frac{3}{2}$

Step 5

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

Interval Notation:

$[\frac{3}{2}, \frac{5}{2}]$

Set-Builder Notation:

${x | \frac{3}{2} \leq x \leq \frac{5}{2}}$

Step 6