Find the Domain and Range y=12/25arccsc(4x-4/7)

$y = \frac{12}{25} arccsc (4 x - \frac{4}{7})$

Set the argument in $arccsc (4 x - \frac{4}{7})$ less than or equal to $- 1$ to find where the expression is defined.

$4 x - \frac{4}{7} \leq - 1$

Solve for $x$ .

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

Add $\frac{4}{7}$ to both sides of the inequality.

$4 x \leq - 1 + \frac{4}{7}$

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$4 x \leq - 1 \cdot \frac{7}{7} + \frac{4}{7}$

Combine $- 1$ and $\frac{7}{7}$ .

$4 x \leq \frac{- 1 \cdot 7}{7} + \frac{4}{7}$

Combine the numerators over the common denominator.

$4 x \leq \frac{- 1 \cdot 7 + 4}{7}$

Simplify the numerator.

Multiply $- 1$ by $7$ .

$4 x \leq \frac{- 7 + 4}{7}$

Add $- 7$ and $4$ .

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

Move the negative in front of the fraction.

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

Divide each term in $4 x \leq - \frac{3}{7}$ by $4$ and simplify.

Divide each term in $4 x \leq - \frac{3}{7}$ by $4$ .

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

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$x \leq - \frac{3}{7} \cdot \frac{1}{4}$

Multiply $- \frac{3}{7} \cdot \frac{1}{4}$ .

Multiply $\frac{1}{4}$ by $\frac{3}{7}$ .

$x \leq - \frac{3}{4 \cdot 7}$

Multiply $4$ by $7$ .

$x \leq - \frac{3}{28}$

Set the argument in $arccsc (4 x - \frac{4}{7})$ greater than or equal to $1$ to find where the expression is defined.

$4 x - \frac{4}{7} \geq 1$

Solve for $x$ .

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

Add $\frac{4}{7}$ to both sides of the inequality.

$4 x \geq 1 + \frac{4}{7}$

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

$4 x \geq \frac{7}{7} + \frac{4}{7}$

Combine the numerators over the common denominator.

$4 x \geq \frac{7 + 4}{7}$

Add $7$ and $4$ .

$4 x \geq \frac{11}{7}$

Divide each term in $4 x \geq \frac{11}{7}$ by $4$ and simplify.

Divide each term in $4 x \geq \frac{11}{7}$ by $4$ .

$\frac{4 x}{4} \geq \frac{\frac{11}{7}}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} \geq \frac{\frac{11}{7}}{4}$

Divide $x$ by $1$ .

$x \geq \frac{\frac{11}{7}}{4}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$x \geq \frac{11}{7} \cdot \frac{1}{4}$

Multiply $\frac{11}{7} \cdot \frac{1}{4}$ .

Multiply $\frac{11}{7}$ by $\frac{1}{4}$ .

$x \geq \frac{11}{7 \cdot 4}$

Multiply $7$ by $4$ .

$x \geq \frac{11}{28}$

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

Interval Notation:

$(- \infty, - \frac{3}{28}] \cup [\frac{11}{28}, \infty)$

Set-Builder Notation:

${x | x \leq - \frac{3}{28}, x \geq \frac{11}{28}}$

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

Interval Notation:

$[- \frac{6 π}{25}, 0) \cup (0, \frac{6 π}{25}]$

Set-Builder Notation:

${y | - \frac{6 π}{25} \leq y \leq \frac{6 π}{25}, y \neq 0}$

Determine the domain and range.

Domain: $(- \infty, - \frac{3}{28}] \cup [\frac{11}{28}, \infty), {x | x \leq - \frac{3}{28}, x \geq \frac{11}{28}}$

Range: $[- \frac{6 π}{25}, 0) \cup (0, \frac{6 π}{25}], {y | - \frac{6 π}{25} \leq y \leq \frac{6 π}{25}, y \neq 0}$

Find the Domain and Range y=12/25arccsc(4x-4/7)

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