Find the Domain and Range (2x+1)/x

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: $(-\mathrm{\infty },0)\cup (0,\mathrm{\infty })$. In set-builder notation, it is: $\{x|x\ne 0\}$.

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: $(-\mathrm{\infty },2)\cup (2,\mathrm{\infty })$. In set-builder notation, it is: $\{y|y\ne 2\}$.

Step 4:

Summarize the domain and range of the function. The domain is: $(-\mathrm{\infty },0)\cup (0,\mathrm{\infty })$ and $\{x|x\ne 0\}$. The range is: $(-\mathrm{\infty },2)\cup (2,\mathrm{\infty })$ and $\{y|y\ne 2\}$.

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, $\{x|x\ne 0\}$ 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.