Find the Domain y=(3x+1)/(x-2)
The problem asks you to determine the domain of the function \(y = \frac{3x+1}{x-2}\). The domain of a function is the set of all possible input values (x-values) for which the function is defined. Specifically, for this rational function, you need to identify all the x-values that the function can take without leading to any undefined or non-permissible operations, such as division by zero.
$y = \frac{3 x + 1}{x - 2}$
Answer
Solution:
Step 1:
Identify the values of $x$ that cause the denominator of $\frac{3x+1}{x-2}$ to be zero by solving the equation $x - 2 = 0$.
Step 2:
Solve for $x$ by adding $2$ to each side of the equation, yielding $x = 2$.
Step 3:
Determine the domain by excluding the value that makes the denominator zero. The domain in different notations is:
- Interval Notation: $(-\infty, 2) \cup (2, \infty)$
- Set-Builder Notation: $\{x | x \neq 2\}$
Knowledge Notes:
The domain of a function is the set of all possible input values (usually represented as $x$) for which the function is defined. For rational functions, which are functions that involve a fraction with polynomials in the numerator and the denominator, the domain is all real numbers except where the denominator is zero, since division by zero is undefined.
To find the domain of the function $y = \frac{3x+1}{x-2}$:
Identify Zero Denominator: We first need to find where the denominator equals zero because these are the values that cannot be included in the domain. This is done by setting the denominator equal to zero and solving for $x$.
Solve the Equation: After setting the denominator to zero, we solve the resulting equation to find the specific values of $x$ that are not allowed.
Express the Domain: Finally, we express the domain in a way that includes all other numbers except the ones found in step 2. This can be done using interval notation or set-builder notation:
Interval Notation: This notation uses intervals to describe subsets of real numbers. In this case, we have two intervals, $(-\infty, 2)$ and $(2, \infty)$, which together represent all real numbers except $2$.
Set-Builder Notation: This notation describes a set by specifying a property that its members must satisfy. Here, the set contains all $x$ such that $x$ is not equal to $2$, written as $\{x | x \neq 2\}$.
Understanding these concepts is essential for correctly determining the domain of a function and for working with rational functions in general.
