Find the Domain and Range (2x+1)/x
The question is asking for the determination of two sets related to the function \( f(x) = \frac{2x + 1}{x} \). Specifically, it requires:
The Domain: This is the set of all possible input values (x-values) that can be used in the function without causing any mathematical inconsistencies or undefined operations. For this specific function, the requester is being asked to identify the values of \( x \) for which the function \( f(x) \) is defined.
The Range: This is the set of all possible output values (y-values) that the function can produce. For the given function, the question asks to determine the values of \( f(x) \) that will result from substituting all the permitted values of \( x \) from the domain into the function.
$\frac{2 x + 1}{x}$
Answer
Solution:
Step 1:
Identify the non-permissible values for $x$ by setting the denominator of $\frac{2x + 1}{x}$ to zero. Solve for $x$: $x = 0$.
Step 2:
The domain consists of all real numbers except the non-permissible value found in step 1. In interval notation, the domain is: $(-\infty, 0) \cup (0, \infty)$. In set-builder notation, it is: $\{x | x \neq 0\}$.
Step 3:
To find the range, analyze the graph of the function or use algebraic methods. The range excludes the horizontal asymptote of the function, which is $y = 2$. In interval notation, the range is: $(-\infty, 2) \cup (2, \infty)$. In set-builder notation, it is: $\{y | y \neq 2\}$.
Step 4:
Summarize the domain and range of the function. The domain is: $(-\infty, 0) \cup (0, \infty)$ and $\{x | x \neq 0\}$. The range is: $(-\infty, 2) \cup (2, \infty)$ and $\{y | y \neq 2\}$.
Knowledge Notes:
The domain of a function is the set of all possible input values (usually represented by $x$) for which the function is defined. For rational functions, any value that makes the denominator zero is excluded from the domain.
The range of a function is the set of all possible output values (usually represented by $y$) that the function can produce. For rational functions, the range can be determined by finding the horizontal asymptote and identifying any restrictions based on the function's behavior.
Interval notation is a way of writing subsets of the real number line. An interval notation consists of a pair of numbers that define the endpoints of the interval, and parentheses or brackets are used to indicate whether the endpoints are included (brackets) or excluded (parentheses).
Set-builder notation is a mathematical notation for describing a set by stating the properties that its members must satisfy. For example, $\{x | x \neq 0\}$ describes the set of all real numbers $x$ such that $x$ is not equal to zero.
In rational functions like $\frac{2x + 1}{x}$, the denominator cannot be zero because division by zero is undefined. Therefore, the value that makes the denominator zero is not part of the domain.
A horizontal asymptote of a function is a horizontal line that the graph of the function approaches as $x$ goes to infinity or negative infinity. The function may approach different horizontal asymptotes as $x$ goes to positive infinity or negative infinity.
