Algebra Examples

$y = 2 \arcsin (12 x + 8)$

Step 1

Set the argument in $\arcsin (12 x + 8)$ greater than or equal to $- 1$ to find where the expression is defined.

$12 x + 8 \geq - 1$

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 $8$ from both sides of the inequality.

$12 x \geq - 1 - 8$

Step 2.1.2

Subtract $8$ from $- 1$ .

$12 x \geq - 9$

Step 2.2

Divide each term in $12 x \geq - 9$ by $12$ and simplify.

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Step 2.2.1

Divide each term in $12 x \geq - 9$ by $12$ .

$\frac{12 x}{12} \geq \frac{- 9}{12}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $12$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{12 x}{12} \geq \frac{- 9}{12}$

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{- 9}{12}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Cancel the common factor of $- 9$ and $12$ .

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Step 2.2.3.1.1

Factor $3$ out of $- 9$ .

$x \geq \frac{3 (- 3)}{12}$

Step 2.2.3.1.2

Cancel the common factors.

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Step 2.2.3.1.2.1

Factor $3$ out of $12$ .

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

Step 2.2.3.1.2.2

Cancel the common factor.

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

Step 2.2.3.1.2.3

Rewrite the expression.

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

Step 2.2.3.2

Move the negative in front of the fraction.

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

Step 3

Set the argument in $\arcsin (12 x + 8)$ less than or equal to $1$ to find where the expression is defined.

$12 x + 8 \leq 1$

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 $8$ from both sides of the inequality.

$12 x \leq 1 - 8$

Step 4.1.2

Subtract $8$ from $1$ .

$12 x \leq - 7$

Step 4.2

Divide each term in $12 x \leq - 7$ by $12$ and simplify.

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Step 4.2.1

Divide each term in $12 x \leq - 7$ by $12$ .

$\frac{12 x}{12} \leq \frac{- 7}{12}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $12$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{12 x}{12} \leq \frac{- 7}{12}$

Step 4.2.2.1.2

Divide $x$ by $1$ .

$x \leq \frac{- 7}{12}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Move the negative in front of the fraction.

$x \leq - \frac{7}{12}$

Step 5

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

Interval Notation:

$[- \frac{3}{4}, - \frac{7}{12}]$

Set-Builder Notation:

${x | - \frac{3}{4} \leq x \leq - \frac{7}{12}}$

Step 6

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

Interval Notation:

$[- π, π]$

Set-Builder Notation:

${y | - π \leq y \leq π}$

Step 7

Determine the domain and range.

Domain: $[- \frac{3}{4}, - \frac{7}{12}], {x | - \frac{3}{4} \leq x \leq - \frac{7}{12}}$

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

Step 8