Find the Domain and Range y=tan(x-pi)
The problem you have provided involves understanding the concepts of domain and range of a trigonometric function, specifically the tangent function. The domain refers to all the possible input values (x-values) for which the function is defined, and the range refers to all possible output values (y-values) that the function can produce. The function you have given is y = tan(x - π), which is a transformation of the basic tangent function (y = tan x) shifted to the right by π units.
To find the domain, one must consider the values of x for which tan(x - π) is defined; in other words, identify the values of x that would not result in an undefined expression (such as a division by zero in the sine/cosine representation of the tangent function).
For the range, it involves determining the set of values that y can take, given the properties of the tangent function. The range of the basic tangent function is all real numbers, due to its periodic nature and the fact that it has no horizontal asymptotes, and one would assess if and how this range is affected by the transformation of shifting the function by π units horizontally.
The question asks for both of these characteristics of the given function to be identified.
$y = tan \left(\right. x - \pi \left.\right)$
Answer
Solution:
Step 1:
Identify the values for which $tan(x - \pi)$ is not defined by setting $x - \pi$ equal to $\frac{\pi}{2} + \pi n$ where $n$ is any integer. Thus, we have $x - \pi = \frac{\pi}{2} + \pi n$.
Step 2:
Isolate $x$ by moving terms that do not include $x$ to the other side of the equation.
Step 2.1:
Add $\pi$ to both sides to get $x = \frac{\pi}{2} + \pi n + \pi$.
Step 2.2:
Express $\pi$ with a common denominator by multiplying by $\frac{2}{2}$, resulting in $x = \pi n + \frac{\pi}{2} + \pi \cdot \frac{2}{2}$.
Step 2.3:
Combine like terms to simplify: $x = \pi n + \frac{\pi}{2} + \frac{2\pi}{2}$.
Step 2.4:
Combine the terms over a common denominator: $x = \pi n + \frac{\pi + 2\pi}{2}$.
Step 2.5:
Sum the terms in the numerator: $x = \pi n + \frac{3\pi}{2}$.
Step 3:
The domain consists of all $x$ values that do not make the expression undefined. In set-builder notation: $\{ x | x \neq \pi n + \frac{3\pi}{2}, n \in \mathbb{Z} \}$.
Step 4:
The range includes all possible $y$ values, which can be determined from the graph of the function. In interval notation, the range is $(-\infty, \infty)$ and in set-builder notation: $\{ y | y \in \mathbb{R} \}$.
Step 5:
Conclude the domain and range of the function:
Domain: $\{ x | x \neq \pi n + \frac{3\pi}{2}, n \in \mathbb{Z} \}$ Range: $(-\infty, \infty)$, $\{ y | y \in \mathbb{R} \}$
Knowledge Notes:
The domain of a function is the set of all possible input values (typically 'x' values) for which the function is defined.
The range of a function is the set of all possible output values (typically 'y' values) that the function can produce.
The tangent function, $\tan(x)$, is undefined at $\frac{\pi}{2} + \pi n$, where $n$ is an integer, because these are the vertical asymptotes of the function.
When finding the domain of trigonometric functions like tangent, we exclude the values where the function is undefined.
The range of the tangent function is all real numbers, which is denoted by $(-\infty, \infty)$ in interval notation or $\{ y | y \in \mathbb{R} \}$ in set-builder notation.
Set-builder notation is a concise way of expressing a set by specifying a property that its members must satisfy.
Interval notation is a way of representing subsets of the real numbers by using intervals.
$\mathbb{R}$ represents the set of all real numbers.
$\mathbb{Z}$ represents the set of all integers (positive, negative, and zero).
