Find the Domain and Range y=csc(x/2)
The question requires you to determine two sets associated with the function y = csc(x/2). The first set is the domain, which is the set of all possible input values x that the function can accept without leading to any undefined or non-real outputs. For the csc function, which is the reciprocal of the sine function, the domain excludes values that would make the sine function zero (since division by zero is undefined).
The second set is the range, which is the set of all possible output values y that the function can produce by plugging in all the allowed input values from the domain. The range of the csc function is determined by knowing that it is the reciprocal of the sine function, which has a range of [-1, 1], and understanding how this range translates for its reciprocal.
The question asks you to apply this knowledge specifically to csc(x/2), which means you must also consider the period and phase shift effects of the argument x/2 on the sine function and thus on the cosecant function.
$y = csc \left(\right. \frac{x}{2} \left.\right)$
Answer
Solution:
Step 1:
Identify the values of $x$ for which $\csc\left(\frac{x}{2}\right)$ is not defined by setting the inside of the csc function to integer multiples of $\pi$: $\frac{x}{2} = \pi n$ where $n$ is any integer.
Step 2:
Find the values of $x$ that cause the function to be undefined.
Step 2.1:
Double each side of the equation to isolate $x$: $2 \cdot \frac{x}{2} = 2 \cdot \pi n$.
Step 2.2:
Simplify the equation.
Step 2.2.1:
Focus on simplifying the left side of the equation.
Step 2.2.1.1:
Eliminate the common factor of $2$: $\cancel{2} \cdot \frac{x}{\cancel{2}} = x$.
Step 2.2.1.1.2:
Express the simplified equation: $x = 2 \pi n$.
Step 2.2.2:
Simplify the right side of the equation.
Step 2.2.2.1:
Remove the parentheses to get the final form: $x = 2 \pi n$.
Step 3:
Define the domain by excluding the values found in Step 2. The domain in set-builder notation is: $\{ x | x \neq 2 \pi n, n \in \mathbb{Z} \}$.
Step 4:
Determine the range by considering the behavior of the cosecant function. The range in interval notation is: $(-\infty, -1] \cup [1, \infty)$. In set-builder notation, the range is: $\{ y | y \leq -1 \text{ or } y \geq 1 \}$.
Step 5:
Combine the results for the domain and range.
Domain: $\{ x | x \neq 2 \pi n, n \in \mathbb{Z} \}$
Range: $(-\infty, -1] \cup [1, \infty)$, $\{ y | y \leq -1 \text{ or } y \geq 1 \}$
Step 6:
Knowledge Notes:
The problem involves finding the domain and range of the function $y = \csc\left(\frac{x}{2}\right)$. The domain of a function consists of all the input values (x-values) for which the function is defined, while the range consists of all the output values (y-values) that the function can take.
For the cosecant function, which is the reciprocal of the sine function, it is undefined whenever the sine function is zero because division by zero is undefined. Therefore, we find the values of $x$ for which $\sin\left(\frac{x}{2}\right) = 0$, which occurs at integer multiples of $\pi$.
The range of the cosecant function is all real numbers except for the interval $(-1, 1)$ because the sine function has a maximum value of 1 and a minimum value of -1, and taking the reciprocal of these values would approach infinity.
In interval notation, the range is expressed as a union of two intervals that cover all the y-values except for those between -1 and 1. In set-builder notation, the range is described using a condition that the y-values must satisfy.
The domain and range are expressed in set-builder notation to explicitly state the conditions that x and y must meet. The domain excludes the values that make the function undefined, while the range includes all values that the function can output, given the domain.
