Precalculus Examples

$f {(x)}^{- 1} = \arcsin (\frac{x - 6}{7})$

Step 1

Set the argument in $\arcsin (\frac{x - 6}{7})$ greater than or equal to $- 1$ to find where the expression is defined.

$\frac{x - 6}{7} \geq - 1$

Step 2

Solve for $x$ .

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Step 2.1

Multiply both sides by $7$ .

$\frac{x - 6}{7} \cdot 7 \geq - 1 \cdot 7$

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 $7$ .

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Step 2.2.1.1.1

Cancel the common factor.

$\frac{x - 6}{7} \cdot 7 \geq - 1 \cdot 7$

Step 2.2.1.1.2

Rewrite the expression.

$x - 6 \geq - 1 \cdot 7$

Step 2.2.2

Simplify the right side.

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Step 2.2.2.1

Multiply $- 1$ by $7$ .

$x - 6 \geq - 7$

Step 2.3

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

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Step 2.3.1

Add $6$ to both sides of the inequality.

$x \geq - 7 + 6$

Step 2.3.2

Add $- 7$ and $6$ .

$x \geq - 1$

Step 3

Set the argument in $\arcsin (\frac{x - 6}{7})$ less than or equal to $1$ to find where the expression is defined.

$\frac{x - 6}{7} \leq 1$

Step 4

Solve for $x$ .

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Step 4.1

Multiply both sides by $7$ .

$\frac{x - 6}{7} \cdot 7 \leq 1 \cdot 7$

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 $7$ .

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Step 4.2.1.1.1

Cancel the common factor.

$\frac{x - 6}{7} \cdot 7 \leq 1 \cdot 7$

Step 4.2.1.1.2

Rewrite the expression.

$x - 6 \leq 1 \cdot 7$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

Multiply $7$ by $1$ .

$x - 6 \leq 7$

Step 4.3

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

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Step 4.3.1

Add $6$ to both sides of the inequality.

$x \leq 7 + 6$

Step 4.3.2

Add $7$ and $6$ .

$x \leq 13$

Step 5

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

Interval Notation:

$[- 1, 13]$

Set-Builder Notation:

${x | - 1 \leq x \leq 13}$

Step 6