Find the Domain and Range y=12arcsec(2x-5)

$y = 12 arcsec (2 x - 5)$

Set the argument in $arcsec (2 x - 5)$ less than or equal to $- 1$ to find where the expression is defined.

$2 x - 5 \leq - 1$

Solve for $x$ .

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

Add $5$ to both sides of the inequality.

$2 x \leq - 1 + 5$

Add $- 1$ and $5$ .

$2 x \leq 4$

Divide each term in $2 x \leq 4$ by $2$ and simplify.

Divide each term in $2 x \leq 4$ by $2$ .

$\frac{2 x}{2} \leq \frac{4}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} \leq \frac{4}{2}$

Divide $x$ by $1$ .

$x \leq \frac{4}{2}$

Simplify the right side.

Divide $4$ by $2$ .

$x \leq 2$

Set the argument in $arcsec (2 x - 5)$ greater than or equal to $1$ to find where the expression is defined.

$2 x - 5 \geq 1$

Solve for $x$ .

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

Add $5$ to both sides of the inequality.

$2 x \geq 1 + 5$

Add $1$ and $5$ .

$2 x \geq 6$

Divide each term in $2 x \geq 6$ by $2$ and simplify.

Divide each term in $2 x \geq 6$ by $2$ .

$\frac{2 x}{2} \geq \frac{6}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} \geq \frac{6}{2}$

Divide $x$ by $1$ .

$x \geq \frac{6}{2}$

Simplify the right side.

Divide $6$ by $2$ .

$x \geq 3$

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

Interval Notation:

$(- \infty, 2] \cup [3, \infty)$

Set-Builder Notation:

${x | x \leq 2, x \geq 3}$

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

Interval Notation:

$[0, 6 π) \cup (6 π, 12 π]$

Set-Builder Notation:

${y | 0 \leq y \leq 12 π, y \neq 6 π}$

Determine the domain and range.

Domain: $(- \infty, 2] \cup [3, \infty), {x | x \leq 2, x \geq 3}$

Range: $[0, 6 π) \cup (6 π, 12 π], {y | 0 \leq y \leq 12 π, y \neq 6 π}$

Find the Domain and Range y=12arcsec(2x-5)

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