Find the Domain and Range g(x)=arcsin(3x+1)

$g (x) = \arcsin (3 x + 1)$

Step 1

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

$3 x + 1 \geq - 1$

Step 2

Solve for $x$ .

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

Subtract $1$ from both sides of the inequality.

$3 x \geq - 1 - 1$

Subtract $1$ from $- 1$ .

$3 x \geq - 2$

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

Divide each term in $3 x \geq - 2$ by $3$ .

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

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

Step 3

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

$3 x + 1 \leq 1$

Step 4

Solve for $x$ .

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

Subtract $1$ from both sides of the inequality.

$3 x \leq 1 - 1$

Subtract $1$ from $1$ .

$3 x \leq 0$

Divide each term in $3 x \leq 0$ by $3$ and simplify.

Divide each term in $3 x \leq 0$ by $3$ .

$\frac{3 x}{3} \leq \frac{0}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} \leq \frac{0}{3}$

Divide $x$ by $1$ .

$x \leq \frac{0}{3}$

Simplify the right side.

Divide $0$ by $3$ .

$x \leq 0$

Step 5

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

Interval Notation:

$[- \frac{2}{3}, 0]$

Set-Builder Notation:

${x | - \frac{2}{3} \leq x \leq 0}$

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: $[- \frac{2}{3}, 0], {x | - \frac{2}{3} \leq x \leq 0}$

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

Step 8

Find the Domain and Range g(x)=arcsin(3x+1)

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