Trigonometry Examples

$f (x) = \arcsin (\frac{x + 4}{3})$

Step 1

Set the argument in $\arcsin (\frac{x + 4}{3})$ greater than or equal to $- 1$ to find where the expression is defined.

$\frac{x + 4}{3} \geq - 1$

Step 2

Solve for $x$ .

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Step 2.1

Multiply both sides by $3$ .

$\frac{x + 4}{3} \cdot 3 \geq - 1 \cdot 3$

Step 2.2

Simplify.

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Step 2.2.1

Simplify the left side.

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Step 2.2.1.1

Cancel the common factor of $3$ .

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Step 2.2.1.1.1

Cancel the common factor.

$\frac{x + 4}{3} \cdot 3 \geq - 1 \cdot 3$

Step 2.2.1.1.2

Rewrite the expression.

$x + 4 \geq - 1 \cdot 3$

Step 2.2.2

Simplify the right side.

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Step 2.2.2.1

Multiply $- 1$ by $3$ .

$x + 4 \geq - 3$

Step 2.3

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

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Step 2.3.1

Subtract $4$ from both sides of the inequality.

$x \geq - 3 - 4$

Step 2.3.2

Subtract $4$ from $- 3$ .

$x \geq - 7$

Step 3

Set the argument in $\arcsin (\frac{x + 4}{3})$ less than or equal to $1$ to find where the expression is defined.

$\frac{x + 4}{3} \leq 1$

Step 4

Solve for $x$ .

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Step 4.1

Multiply both sides by $3$ .

$\frac{x + 4}{3} \cdot 3 \leq 1 \cdot 3$

Step 4.2

Simplify.

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Step 4.2.1

Simplify the left side.

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Step 4.2.1.1

Cancel the common factor of $3$ .

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Step 4.2.1.1.1

Cancel the common factor.

$\frac{x + 4}{3} \cdot 3 \leq 1 \cdot 3$

Step 4.2.1.1.2

Rewrite the expression.

$x + 4 \leq 1 \cdot 3$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

Multiply $3$ by $1$ .

$x + 4 \leq 3$

Step 4.3

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

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Step 4.3.1

Subtract $4$ from both sides of the inequality.

$x \leq 3 - 4$

Step 4.3.2

Subtract $4$ from $3$ .

$x \leq - 1$

Step 5

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

Interval Notation:

$[- 7, - 1]$

Set-Builder Notation:

${x | - 7 \leq x \leq - 1}$

Step 6