Find the Domain and Range y=csc(1/2x)
The problem involves determining the domain and range of the function y = csc(1/2x). The domain refers to the set of all possible values of the independent variable x for which the function is defined. The range refers to the set of all possible values that the function can take as y when x varies throughout the domain.
The function y = csc(1/2x) is the cosecant function, which is the reciprocal of the sine function. To find the domain, you must identify all x-values where the function makes sense and does not cause any mathematical errors (such as division by zero). For the range, you will need to determine the possible y-values that result from those valid x-values.
In this context, the function has a periodic behavior due to the trigonometric component, and the constraints on the domain and range are related to the properties of the sine function, specifically where it is equal to zero since the cosecant function is undefined at those points.
$y = csc \left(\right. \frac{1}{2} x \left.\right)$
Answer
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: $\{x | x \neq 2k\pi\}$, for any integer $k$
Step:4
Identify the range by considering the values that $y$ can take. Interval Notation: $(-\infty, -1] \cup [1, \infty)$ Set-Builder Notation: $\{y | y \leq -1, y \geq 1\}$
Step:5
Summarize the domain and range. Domain: $\{x | x \neq 2k\pi\}$, for any integer $k$ Range: $(-\infty, -1] \cup [1, \infty), \{y | y \leq -1, y \geq 1\}$
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.
