Problem 17 - Elementary Algebra

Question

Find the domain of the equations. Assume that the independent variable is the variable that appears in the expression to the right of the equal sign. $$ k=\frac{4 t^{2}}{t-1} $$

Step-by-Step Solution

Answer

Answer: The domain of the equation is all real numbers except \(t \neq 1\).

Step 1: Identify the denominator of the equation

The equation is given by: $$ k=\frac{4 t^{2}}{t-1} $$ The denominator of the equation is \((t-1)\).

Step 2: Set the denominator to be different from zero

For the equation to be defined, the denominator should not be equal to zero. So, we need to find the values of \(t\) for which \((t-1) \neq 0\).

Step 3: Solve the inequality

Now, solve the inequality: $$ t - 1 \neq 0 $$ Add 1 to both sides of the inequality: $$ t \neq 1 $$

Step 4: Find the domain

The domain of the equation is the set of all possible values of \(t\) for which the equation is defined. Since the only value that makes the equation undefined is \(t=1\), the domain of the equation is all real numbers except \(t \neq 1\).

Key Concepts

Algebraic Expressions

Understanding algebraic expressions is central to grasping the nature of algebra and finding the domain of an equation. An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like x or y), and operators (like add, subtract, multiply, and divide). These expressions can represent quantities that are unknown or that can vary.

For example, in the equation from the exercise, \( k=\frac{4t^{2}}{t-1} \), the algebraic expression is \(\frac{4t^{2}}{t-1}\). This expression consists of a numerator, which is \((4t^{2})\), and a denominator, which is \((t-1)\). In order to explore the domain of this equation, one must understand how to interpret and work with these components and their implications on the value of the variable.

Inequalities

Inequalities are statements about the relative size or order of two objects, and they are typically presented using symbols such as \<, \leq, \>, \geq\, and \eq\. In the context of finding the domain of an equation, we often use inequalities to determine the values that a variable can or cannot take.

In the provided exercise, we encounter an inequality in Step 3 of the solution: \(t - 1 \eq 0\). This inequality tells us that \(t\) cannot equal 1, since 1 would make the denominator zero and the expression undefined. Thus, we state that the domain is all real numbers except for \(t = 1\), ensuring we do not include any value in the domain that would make the denominator zero or the expression undefined.

Rational Functions

A rational function is another key concept related to the domain of an equation. It is any function that can be expressed as the ratio of two polynomial functions. In its simplest form, a rational function looks like \(f(x) = \frac{p(x)}{q(x)}\), where \(p(x)\) and \(q(x)\) are polynomials and \(q(x) \eq 0\). The most important aspect of a rational function in determining its domain involves ensuring that the denominator never equals zero.

In our exercise, the given equation \(k=\frac{4t^{2}}{t-1}\) is a rational function. As we saw in the solution steps, identifying values that make the denominator zero (\(t = 1\) in this case) is crucial. This single value defines the restriction in the domain of this rational function, representing all real numbers excluding \(t = 1\).