Problem 7
Question
Find the domain of each of the following rational expressions. $$ \frac{k^{2}-4}{5} $$
Step-by-Step Solution
Verified Answer
Expression: $$
\frac{k^{2} - 2k + 4}{5}
$$
Answer: The domain of the given expression is all real numbers, which is represented as $(-\infty, \infty)$.
1Step 1: Analyze the denominator
To find the domain, we should ensure that the denominator is not equal to zero. In this case, the denominator is 5.
2Step 2: Check if the denominator can equal zero
Since 5 is a constant and never equal to zero, there is no need to solve for any constraints on the variable k.
3Step 3: Identify the domain
Since there are no limitations on the variable k due to the denominator, the domain of the given rational expression is all real numbers.
The domain of the given expression is:
$$
(-\infty, \infty)
$$
Key Concepts
Algebraic DomainRational FunctionReal Numbers
Algebraic Domain
When we talk about algebraic domain, we are referring to the set of all possible values that a variable can take on without causing a mathematical expression to become undefined or problematic. In the context of rational expressions, the domain typically excludes values that would make the denominator equal to zero, since division by zero is undefined.
In our example, the expression \(\frac{k^2-4}{5}\) has a constant denominator of 5. As 5 is a non-zero number, it does not impose any restrictions on the variable \(k\), meaning \(k\) can be any real number. Therefore, the domain is all real numbers, which is represented as \( (-\infty, \infty) \). It's essential to remember that the domain must be considered whenever working with an expression, to ensure that the mathematics remains valid and meaningful.
In our example, the expression \(\frac{k^2-4}{5}\) has a constant denominator of 5. As 5 is a non-zero number, it does not impose any restrictions on the variable \(k\), meaning \(k\) can be any real number. Therefore, the domain is all real numbers, which is represented as \( (-\infty, \infty) \). It's essential to remember that the domain must be considered whenever working with an expression, to ensure that the mathematics remains valid and meaningful.
Rational Function
A rational function is a type of expression represented by the ratio of two polynomials, where the numerator and the denominator are both polynomials, and the denominator is not equal to zero. The general form of a rational function is \(\frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomials.
In the exercise above, the given expression \(\frac{k^2-4}{5}\) is a simple rational function, with the numerator \(k^2-4\) being a polynomial of degree 2 and the denominator being the constant 5. Rational functions can have more complex forms with varying degrees and multiple variables, but the key takeaway is that their domains exclude values where the denominator equals zero, ensuring the function is properly defined.
In the exercise above, the given expression \(\frac{k^2-4}{5}\) is a simple rational function, with the numerator \(k^2-4\) being a polynomial of degree 2 and the denominator being the constant 5. Rational functions can have more complex forms with varying degrees and multiple variables, but the key takeaway is that their domains exclude values where the denominator equals zero, ensuring the function is properly defined.
Real Numbers
The set of real numbers includes all the numbers on the number line, encompassing all the rational and irrational numbers. Rational numbers include integers, fractions, and finite decimals, while irrational numbers are those that cannot be expressed as a simple fraction, such as \(\pi\) and \(\sqrt{2}\).
The domain of the given rational expression includes all real numbers, indicating there are no restrictions on the value that \(k\) can take. However, not all algebraic expressions have such an unrestricted domain. For example, if the denominator of a rational function were \(k-3\), the domain would exclude the number 3 to prevent division by zero. Recognizing this broad category of numbers is essential for understanding where rational functions are defined and how to properly analyze their behavior and restrictions.
The domain of the given rational expression includes all real numbers, indicating there are no restrictions on the value that \(k\) can take. However, not all algebraic expressions have such an unrestricted domain. For example, if the denominator of a rational function were \(k-3\), the domain would exclude the number 3 to prevent division by zero. Recognizing this broad category of numbers is essential for understanding where rational functions are defined and how to properly analyze their behavior and restrictions.
Other exercises in this chapter
Problem 7
For the following problems, solve the rational equations. $$ \frac{54}{y}=\frac{27}{4} $$
View solution Problem 7
Perform each division. $$ \frac{x^{2}-4}{x^{2}+x-6} \div \frac{x^{2}+x-2}{x^{2}+4 x+3} $$
View solution Problem 7
Add or Subtract the following rational expressions. $$ \frac{4 x+3}{x^{2}-x-6}-\frac{8 x-4}{(x+2)(x-3)} $$
View solution Problem 7
Reduce each of the following fractions to lowest terms. $$ \frac{2 x-y}{y-2 x} $$
View solution