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Function analysis helps us understand the behavior and properties of a mathematical function. When analyzing functions such as \( G(t) = \frac{2}{t^2 - 16} \), we focus on identifying restrictions and key characteristics. One primary aspect is the **domain**, which represents all possible inputs. For \( G(t) \), we determine that the function is undefined when the denominator equals zero. By solving \( t^2 - 16 = 0 \), we find the function is undefined at \( t = 4 \) and \( t = -4 \). This indicates that the domain excludes these values:

• Domain: \( (-\infty, -4) \cup (-4, 4) \cup (4, \infty) \)

Understanding the **range** is equally vital. It describes all possible outputs. Here, as \( t \) approaches values causing the denominator to zero, \( G(t) \) tends towards positive or negative infinity, excluding zero. Thus, the range is all real numbers except zero:

• Range: \( (-\infty, 0) \cup (0, \infty) \)

Asymptotes play a crucial role in understanding functions, especially when analyzing rational functions like \( G(t) = \frac{2}{t^2 - 16} \). An **asymptote** is a line that a function approaches but does not touch or cross. In this function, vertical asymptotes occur where the function is undefined—at \( t = 4 \) and \( t = -4 \). These are identified by setting the denominator \( t^2 - 16 \) to zero. The function nears infinity or negative infinity close to these points, making

• Vertical asymptotes: \( t = 4 \) and \( t = -4 \)

As for horizontal asymptotes, they indicate what happens to \( G(t) \) as \( t \) approaches infinity or negative infinity. Since the degree of the denominator is greater than the numerator's degree, the function approaches zero as \( t \) moves towards infinity:

• Horizontal asymptote: \( y = 0 \)

Rational functions are expressions of the form \( \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials, typically with \( Q(x) \) not equal to zero. The function \( G(t) = \frac{2}{t^2 - 16} \) is a classic example, where the numerator \( P(t) \) is 2 and the denominator \( Q(t) \) is \( t^2 - 16 \). This function type often features:

• **Domain restrictions**: Determined by the values that make the denominator zero, hence the exclusions \( t eq 4 \) and \( t eq -4 \).
• **Discontinuities**: Places where the function is not continuous due to undefined points.
• **End behavior**: Driven by the function's degree and leads to asymptotic behavior, often described by horizontal and oblique asymptotes.

Rational functions like \( G(t) \) smoothly articulate how they behave across their domain, excluding points leading to undefined terms. With a thorough understanding of these characteristics, analyzing and predicting function behavior becomes more intuitive and straightforward.