Problem 26 - Intermediate Algebra

Question

Specify the domain for each of the functions. $$f(t)=\frac{-2 t}{t^{2}-25}$$

Step-by-Step Solution

Answer

Domain: \((-\infty, -5) \cup (-5, 5) \cup (5, \infty)\).

Step 1: Identify the Function Type

The function given is a rational function, which is written as the ratio of two polynomials. Specifically, we have the numerator \[-2t\]and the denominator \[t^2 - 25.\]

Step 2: Determine Restrictions from the Denominator

For the domain of a rational function, the denominator cannot be zero, as division by zero is undefined. So, we set the denominator equal to zero and solve for \(t\):\[t^2 - 25 = 0.\]

Step 3: Solve the Equation from Step 2

Solve for \(t\) by factoring or other algebraic methods:\[t^2 - 25 = (t - 5)(t + 5) = 0.\]Setting each factor equal to zero, we find:\[t - 5 = 0 \Rightarrow t = 5\]and \[t + 5 = 0 \Rightarrow t = -5.\]

Step 4: Write the Domain

The values \(t = 5\) and \(t = -5\) are points where the denominator becomes zero; therefore, these values are removed from the domain. The domain of \(f(t)\) is all real numbers except \(t = 5\) and \(t = -5\). In interval notation, the domain is:\((-\infty, -5) \cup (-5, 5) \cup (5, \infty).\)

Key Concepts

Rational Functions

Rational functions are an important part of algebra, and they involve the division of one polynomial by another. The function given in the problem is an example of a rational function. It is expressed as \[ f(t) = \frac{-2t}{t^{2} - 25} \]where the numerator is -2tand the denominator is\( t^2 - 25 \). Understanding rational functions involves looking at both the numerator and the denominator, particularly focusing on when the denominator might become zero, since division by zero is undefined. If we identify any value that makes the denominator zero, those are the values that are not part of the domain, i.e., they are not allowed in the function.

Polynomials

Polynomials are expressions that involve terms composed of variables and coefficients, raised to whole number powers. In the function \( f(t) = \frac{-2t}{t^{2} - 25} \), both -2tand\( t^2 - 25 \)are polynomials. The numerator, -2t,is a first-degree polynomial, while the denominator,\( t^2 - 25 \),is a second-degree polynomial. The structure and degree of these polynomials help in identifying the kind of algebraic manipulations we can perform, such as factoring. In this case, \( t^2 - 25 \) is factored as \((t - 5)(t + 5)\),which is key to determining where the denominator equals zero.

Interval Notation

Interval notation is a method used to describe the set of all numbers between two endpoints. This is particularly helpful in representing domains for functions with certain restrictions. For the function \( f(t) = \frac{-2t}{t^{2} - 25} \), we exclude the points \( t = 5 \) and \( t = -5 \) from the domain because they make the denominator zero. Thus, in interval notation, we express the domain as \((-\infty,-5) \, \cup \, (-5,5) \, \cup \, (5, \infty)\).This notation includes all the real numbers except for \( -5 \) and \( 5 \). The parentheses in the intervals indicate that the endpoints are not included in the domain, reflecting the fact that these values result in an undefined expression.