Problem 23 - Intermediate Algebra

Question

Specify the domain for each of the functions. $$f(t)=\frac{4}{t^{2}+9}$$

Step-by-Step Solution

Answer

The domain of \( f(t) = \frac{4}{t^2 + 9} \) is all real numbers \( t \).

Step 1: Understanding the Function

The given function is \( f(t) = \frac{4}{t^2 + 9} \). It is a rational function, which are defined everywhere the denominator is not equal to zero.

Step 2: Analyzing the Denominator

For the function to be defined, the denominator \( t^2 + 9 \) must not be zero. So, we need to check where \( t^2 + 9 eq 0 \).

Step 3: Solving the Condition

Since \( t^2 \) is always non-negative for real numbers and 9 is positive, \( t^2 + 9 \) is always greater than zero for all real \( t \). It will never be zero because the smallest value of \( t^2 \) is zero, making \( t^2 + 9 = 9 \).

Step 4: Conclusion on Domain

As the expression \( t^2 + 9 \) never reaches zero for any real number, there are no restrictions on \( t \). Therefore, the domain of \( f(t) \) is all real numbers.

Key Concepts

Rational Functions

Rational functions are the quotient of two polynomials. The general format is \[ f(x) = \frac{P(x)}{Q(x)} \] where both \( P(x) \) and \( Q(x) \) are polynomials. The key feature of rational functions is that they are undefined wherever the denominator, \( Q(x) \), equals zero because division by zero is not possible.

• Numerator: Can be any polynomial, including constants.
• Denominator: Must be non-zero to ensure the function is defined.

A crucial part of working with rational functions is finding their domain, which involves determining where the denominator does not equal zero. This helps in understanding where the function is defined and can be evaluated.

Function Domain

The domain of a function is the set of all input values for which the function is defined. In the case of rational functions, the domain excludes values that make the denominator zero.

• Identify the denominator: Look at the function and locate the denominator.
• Solve for zero: Set the denominator equal to zero and solve for the variable to find forbidden values.
• Exclude these from the domain: Any value that makes the denominator zero must be excluded from the domain.

In the exercise given, the function \( f(t) = \frac{4}{t^2 + 9} \) has a denominator \( t^2 + 9 \). Since this quadratic expression is always positive for real values of \( t \), there is no need to exclude any values. Therefore, the domain here consists of all real numbers.

Real Numbers

Real numbers include all possible values on the number line. These encompass the negatives, positives, and zero. Real numbers are denoted by the symbol \( \mathbb{R} \), representing a continuous set with no breaks or gaps.

• Integers: Whole numbers, both positive and negative, including zero.
• Rational numbers: Numbers that can be expressed as a fraction \( \frac{a}{b} \), where \( a \) and \( b \) are integers, and \( b eq 0 \).
• Irrational numbers: Numbers that cannot be expressed as fractions and have non-repeating, non-terminating decimal expansions (like \( \pi \) or \( \sqrt{2} \)).

Understanding real numbers helps in grasping the concept of function domains, as domains are often specified in terms of which real numbers can be inputs to functions. In our case, all real numbers are valid inputs to the function because the denominator \( t^2 + 9 \) is always positive, thereby not restricting \( t \).