Find the Domain and Range y=6/11arcsin((2x)/5+3)

$y = \frac{6}{11} \arcsin (\frac{2 x}{5} + 3)$

Step 1

Set the argument in $\arcsin (\frac{2 x}{5} + 3)$ greater than or equal to $- 1$ to find where the expression is defined.

$\frac{2 x}{5} + 3 \geq - 1$

Step 2

Solve for $x$ .

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

Subtract $3$ from both sides of the inequality.

$\frac{2 x}{5} \geq - 1 - 3$

Subtract $3$ from $- 1$ .

$\frac{2 x}{5} \geq - 4$

Multiply both sides by $5$ .

$\frac{2 x}{5} \cdot 5 \geq - 4 \cdot 5$

Simplify.

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{2 x}{5} \cdot 5 \geq - 4 \cdot 5$

Rewrite the expression.

$2 x \geq - 4 \cdot 5$

Simplify the right side.

Multiply $- 4$ by $5$ .

$2 x \geq - 20$

Divide each term in $2 x \geq - 20$ by $2$ and simplify.

Divide each term in $2 x \geq - 20$ by $2$ .

$\frac{2 x}{2} \geq \frac{- 20}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} \geq \frac{- 20}{2}$

Divide $x$ by $1$ .

$x \geq \frac{- 20}{2}$

Simplify the right side.

Divide $- 20$ by $2$ .

$x \geq - 10$

Step 3

Set the argument in $\arcsin (\frac{2 x}{5} + 3)$ less than or equal to $1$ to find where the expression is defined.

$\frac{2 x}{5} + 3 \leq 1$

Step 4

Solve for $x$ .

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

Subtract $3$ from both sides of the inequality.

$\frac{2 x}{5} \leq 1 - 3$

Subtract $3$ from $1$ .

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

Multiply both sides by $5$ .

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

Simplify.

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

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

Rewrite the expression.

$2 x \leq - 2 \cdot 5$

Simplify the right side.

Multiply $- 2$ by $5$ .

$2 x \leq - 10$

Divide each term in $2 x \leq - 10$ by $2$ and simplify.

Divide each term in $2 x \leq - 10$ by $2$ .

$\frac{2 x}{2} \leq \frac{- 10}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} \leq \frac{- 10}{2}$

Divide $x$ by $1$ .

$x \leq \frac{- 10}{2}$

Simplify the right side.

Divide $- 10$ by $2$ .

$x \leq - 5$

Step 5

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

Interval Notation:

$[- 10, - 5]$

Set-Builder Notation:

${x | - 10 \leq x \leq - 5}$

Step 6

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

Interval Notation:

$[- \frac{3 π}{11}, \frac{3 π}{11}]$

Set-Builder Notation:

${y | - \frac{3 π}{11} \leq y \leq \frac{3 π}{11}}$

Step 7

Determine the domain and range.

Domain: $[- 10, - 5], {x | - 10 \leq x \leq - 5}$

Range: $[- \frac{3 π}{11}, \frac{3 π}{11}], {y | - \frac{3 π}{11} \leq y \leq \frac{3 π}{11}}$

Step 8

Find the Domain and Range y=6/11arcsin((2x)/5+3)

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