Problem 6 - Thomas Calculus

Question

In Exercises \(1-6,\) find the domain and range of each function. $$G(t)=\frac{2}{t^{2}-16}$$

Step-by-Step Solution

Answer

Domain: \((-\infty, -4) \cup (-4, 4) \cup (4, \infty)\) Range: \((-\infty, 0) \cup (0, \infty)\)

Step 1: Identify Domain Restrictions

The domain of a function includes all possible input values (in this case, values of \(t\)) that do not cause the function to be undefined. For the function \(G(t) = \frac{2}{t^2 - 16}\), the expression \(t^2 - 16\) cannot equal zero because division by zero is undefined.

Step 2: Solve for Domain Exclusions

To find where \(t^2 - 16 = 0\), solve the equation:\[t^2 - 16 = 0 \t^2 = 16 \t = \pm 4\]Thus, \(t\) cannot be \(4\) or \(-4\).

Step 3: State the Domain

The domain of \(G(t)\) is all real numbers except \(t = 4\) and \(t = -4\). In interval notation, this is written as: \[(-\infty, -4) \cup (-4, 4) \cup (4, \infty)\]

Step 4: Identify the Range

Since the numerator of \(G(t)\) is a constant \(2\) and the denominator \(t^2 - 16\) is quadratic, pay attention to how the sign of \(t^2 - 16\) affects the function's output. \(t^2 - 16\) can be any non-zero value, which makes \(\frac{2}{t^2 - 16}\) take all non-zero real numbers.

Step 5: State the Range

Because the function can approach any real number except zero, the range of \(G(t)\) is all real numbers except zero. In interval notation, the range is written as: \[(-\infty, 0) \cup (0, \infty)\]

Key Concepts

Domain and Range

When we talk about the domain of a function, we're focusing on all the values that the input (like the variable \( t \)) can take. To find the domain, we need to ensure that the function is defined at those values. For a rational function like \( G(t) = \frac{2}{t^2 - 16} \), the denominator \( t^2 - 16 \) cannot be zero because division by zero is undefined.

We solve \( t^2 - 16 = 0 \) to find which values to exclude. By solving, we find \( t = 4 \) and \( t = -4 \), and thus, these values are not in the domain. Therefore, the domain is all real numbers except these two points:

• Interval notation: \( (-\infty, -4) \cup (-4, 4) \cup (4, \infty) \)

Next, the range describes all possible outputs of the function, which is about the values that the function can achieve. For \( G(t) \), the range excludes 0 because \( \frac{2}{t^2-16} \) can't be zero for any real number \( t \) when \( t^2-16 e 0 \). Hence, the range is:

• Interval notation: \( (-\infty, 0) \cup (0, \infty) \)

Rational Functions

Rational functions are expressions that incorporate one polynomial divided by another. In the function \( G(t) = \frac{2}{t^2 - 16} \), the numerator \( 2 \) is a constant polynomial, and the denominator \( t^2 - 16 \) is a quadratic polynomial.

An important feature of rational functions is that they are undefined where the denominator equals zero. Thus, the domain is restricted around this fact. To find these restrictions, we set the denominator equal to zero and solve for the variable. In this case, \( t^2 = 16 \), giving us \( t = \pm 4 \). Therefore, \( t = 4 \) and \( t = -4 \) make the function undefined, acting as vertical asymptotes where the behavior of the function is discontinuous.

One should note how the constant numerator \( 2 \) affects the function's behavior because it means the function never actually becomes zero. As a result, the range of \( G(t) \) includes all non-zero numbers, influencing how we determine the function's range.

Interval Notation

Interval notation is a concise way to express sets of numbers, particularly useful for describing domains and ranges in calculus functions. For \( G(t) = \frac{2}{t^2-16} \), interval notation helps us clearly express which values \( t \) can and can't take.

In interval notation:

• Parentheses \( ( \) or \( ) \) indicate that the endpoint is not included in the interval.
• Brackets \( [ \) or \( ] \) would include the endpoint in the interval, but these are not used here because \( t = 4 \) and \( t = -4 \) are not part of the domain.

The domain, excluding \( t = 4 \) and \( t = -4 \), is written as \((-\infty, -4) \cup (-4, 4) \cup (4, \infty)\). Similarly, the range, which excludes zero, is expressed by \((-\infty, 0) \cup (0, \infty)\). Both of these use union (\( \cup \)) to join intervals that aren't continuous. Understanding interval notation is key to clearly and effectively representing these sets of numbers.