Find the Domain and Range y=arccsc(5x+3/2)

$y = arccsc (5 x + \frac{3}{2})$

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

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

Solve for $x$ .

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

Subtract $\frac{3}{2}$ from both sides of the inequality.

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

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Combine $- 1$ and $\frac{2}{2}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $- 1$ by $2$ .

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

Subtract $3$ from $- 2$ .

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

Move the negative in front of the fraction.

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

Divide each term in $5 x \leq - \frac{5}{2}$ by $5$ and simplify.

Divide each term in $5 x \leq - \frac{5}{2}$ by $5$ .

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

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

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

Cancel the common factor of $5$ .

Move the leading negative in $- \frac{5}{2}$ into the numerator.

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

Factor $5$ out of $- 5$ .

$x \leq \frac{5 (- 1)}{2} \cdot \frac{1}{5}$

Cancel the common factor.

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

Rewrite the expression.

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

Move the negative in front of the fraction.

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

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

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

Solve for $x$ .

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

Subtract $\frac{3}{2}$ from both sides of the inequality.

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

Write $1$ as a fraction with a common denominator.

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

Combine the numerators over the common denominator.

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

Subtract $3$ from $2$ .

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

Move the negative in front of the fraction.

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

Divide each term in $5 x \geq - \frac{1}{2}$ by $5$ and simplify.

Divide each term in $5 x \geq - \frac{1}{2}$ by $5$ .

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

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

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

Multiply $- \frac{1}{2} \cdot \frac{1}{5}$ .

Multiply $\frac{1}{5}$ by $\frac{1}{2}$ .

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

Multiply $5$ by $2$ .

$x \geq - \frac{1}{10}$

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

Interval Notation:

$(- \infty, - \frac{1}{2}] \cup [- \frac{1}{10}, \infty)$

Set-Builder Notation:

${x | x \leq - \frac{1}{2}, x \geq - \frac{1}{10}}$

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

Interval Notation:

$[- \frac{π}{2}, 0) \cup (0, \frac{π}{2}]$

Set-Builder Notation:

${y | - \frac{π}{2} \leq y \leq \frac{π}{2}, y \neq 0}$

Determine the domain and range.

Domain: $(- \infty, - \frac{1}{2}] \cup [- \frac{1}{10}, \infty), {x | x \leq - \frac{1}{2}, x \geq - \frac{1}{10}}$

Range: $[- \frac{π}{2}, 0) \cup (0, \frac{π}{2}], {y | - \frac{π}{2} \leq y \leq \frac{π}{2}, y \neq 0}$

Find the Domain and Range y=arccsc(5x+3/2)

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