Find the Domain and Range y=csc(1/2x)

Solution:

Step:1

Identify the values that make $\csc \left(\frac{1}{2}x\right)$ undefined by setting the inside of the cosecant function to $k\pi$, where $k$ is an integer. $\frac{1}{2}x=k\pi$

Step:2

Determine the values of $x$.

Step:2.1

Double each side of the equation to isolate $x$. $2\cdot \left(\frac{1}{2}x\right)=2\cdot (k\pi )$

Step:2.2

Proceed to simplify the equation.

Step:2.2.1

Focus on the left-hand side.

Step:2.2.1.1

Combine the terms $2\cdot \left(\frac{1}{2}x\right)$.

Step:2.2.1.1.1

Merge the $2$ and $\frac{1}{2}$ with $x$. $2\cdot \frac{x}{2}=2\cdot (k\pi )$

Step:2.2.1.1.2

Eliminate the common factor of $2$.

Step:2.2.1.1.2.1

Remove the common factor. $\cancel{2}\cdot \frac{x}{\cancel{2}}=2\cdot (k\pi )$

Step:2.2.1.1.2.2

Rewrite the simplified expression. $x=2\cdot (k\pi )$

Step:2.2.2

Now, simplify the right-hand side.

Step:2.2.2.1

Discard the parentheses. $x=2k\pi$

Step:3

Define the domain as all real numbers except where $x$ makes $\csc \left(\frac{1}{2}x\right)$ undefined. Set-Builder Notation: $\{x|x\ne 2k\pi \}$, for any integer $k$

Step:4

Identify the range by considering the values that $y$ can take. Interval Notation: $(-\mathrm{\infty },-1]\cup [1,\mathrm{\infty })$ Set-Builder Notation: $\{y|y\le -1,y\ge 1\}$

Step:5

Summarize the domain and range. Domain: $\{x|x\ne 2k\pi \}$, for any integer $k$ Range: $(-\mathrm{\infty },-1]\cup [1,\mathrm{\infty }),\{y|y\le -1,y\ge 1\}$

Step:6

Knowledge Notes:

The domain of a function consists of all the input values (x-values) for which the function is defined. For the cosecant function, $\csc (\theta )$, the function is undefined when $\theta$ is an integer multiple of $\pi$, since cosecant is the reciprocal of sine and sine is zero at these points.

The range of a function is the set of all possible output values (y-values) it can produce. For the cosecant function, which is the reciprocal of the sine function, the range consists of all real numbers except those between -1 and 1, as the sine function has a maximum value of 1 and a minimum value of -1, and taking the reciprocal of values between -1 and 1 would result in values not defined for the cosecant function.

In interval notation, the domain and range are expressed as intervals of real numbers. Set-builder notation expresses the domain and range in terms of a condition that the elements must satisfy.

To find the domain and range of $y=\csc \left(\frac{1}{2}x\right)$, we first determine where the function is undefined, which is where the inside of the cosecant function is an integer multiple of $\pi$. The domain is then all real numbers except these points. The range is determined by the nature of the cosecant function, which can take on any real value greater than or equal to 1 or less than or equal to -1.