Problem 9 - Basic Technical Mathematics with Calculus

Question

Find the domain and range of the given functions. $$f(s)=\frac{2}{s^{2}}$$

Step-by-Step Solution

Answer

Domain: \((-1e, 0) \cup (0, 1e)\), Range: \((0, 1e)\).

Step 1: Identify Values That Make the Denominator Zero

The function is undefined where the denominator is zero, as division by zero is not possible. The denominator here is \(s^2\). Set it to zero and solve for \(s\): \(s^2 = 0\). This gives \(s = 0\).

Step 2: Establish the Domain of the Function

The domain of \(f(s) = \frac{2}{s^2}\) includes all real numbers except where the function is undefined. From Step 1, \(s = 0\) causes the denominator to be zero. Therefore, the domain is all real numbers except zero, or \((-1e, 0) \cup (0, 1e)\).

Step 3: Identify the Range of the Function

The range of \(f(s)\) is determined by the possible output values \(f(s)\) can produce. Since \(f(s) = \frac{2}{s^2}\), as \(s^2\) becomes very large or very small, \(f(s)\) approaches zero. However, \(f(s)\) is never zero because \(2/s^2\) is always positive, making the range \((0, 1e)\).

Key Concepts

Denominator Zero

When a denominator in a function equals zero, it causes the function to become undefined at that particular point. This is because division by zero is mathematically impossible and leads to an operation that doesn’t yield a real number result. To resolve functions like these, it is essential to identify values that make the denominator zero and exclude them from the domain.
In the function given, \(f(s) = \frac{2}{s^2}\), the denominator is \(s^2\). Setting \(s^2 = 0\) and solving for \(s\) gives \(s = 0\).

• This means that when \(s\) is zero, the denominator is zero and the function is undefined.
• Thus, the value \(s = 0\) must be excluded from the domain of the function.

Understanding this concept is crucial for ensuring that functions do not become undefined due to a zero denominator.

Real Numbers

Real numbers include all possible numbers along the continuous number line, without any gaps. They incorporate both rational numbers (like integers and fractions) and irrational numbers (like the square root of non-square-perfects).
In the context of function domains, a real number set refers to all numbers excluding the ones that cause issues such as making a denominator zero. For the function \(f(s) = \frac{2}{s^2}\), since \(s = 0\) is not allowed, the domain is all real numbers except zero.

• This is expressed as: \((- \infty, 0) \cup (0, \infty)\).
• It means the domain doesn't include zero but includes every other real number.

Understanding this helps identify the domain of functions that otherwise span all real numbers.

Undefined Function

A function becomes undefined in situations where typical mathematical operations cannot be performed. One common cause of an undefined function is when attempting to divide by zero.
For \(f(s) = \frac{2}{s^2}\), the denominator \(s^2\) becomes zero when \(s = 0\), hence making the function undefined at this point. The solution involves removing such points where the function is undefined from its domain.

• However, excluding these points ensures that the function remains valid wherever it is defined.
• Even though the function is undefined at certain points, the range of the function might still span a continuous set depending on how the function behaves elsewhere.
• For example, \(f(s)\) produces values that are always positive when \(s eq 0\), making the range \((0, \infty)\).

These considerations are important when determining which values are part of a function's domain and range.