Question

In Exercises \(7-20,\) find the vertical asymptotes (if any) of the function. $$ g(t)=\frac{t-1}{t^{2}+1} $$

Short answer

The function \(g(t)=\frac{t-1}{t^{2}+1}\) has no vertical asymptotes.

Step-by-step solution

Find where the denominator equals zero

To find the vertical asymptotes, first find the values of \(t\) that make the denominator zero. Solve the equation \(t^{2}+1 = 0\).

Analyze the results

The equation \(t^2 +1 =0\) has no real solutions since \(t^2\) is always non-negative for every real number \(t\), and so \(t^2 + 1\) is always positive. Therefore, the denominator of the function never equals zero.

Conclusion

Since the denominator never equals zero, the function \(g(t)=\frac{t-1}{t^{2}+1}\) has no vertical asymptotes.

Key concepts

Calculus

Calculus is an essential mathematical tool used in various scientific fields, including engineering, physics, and economics. It focuses on the study of change and motion and is divided into two main branches: differential calculus and integral calculus. Differential calculus is concerned with the concept of a derivative, which examines how functions change over time, while integral calculus explores the concept of an integral, which measures the accumulation of quantities over time.

When analyzing functions in calculus, we're often interested in understanding their behavior as variables approach certain points or infinity. This includes identifying points of discontinuity and the tendency of functions as they get close to a particular value, which leads to the concept of limits. Limits help us understand the behavior of functions at points that might not be clearly defined by the function's equation.

Rational Function Analysis

Rational function analysis is a part of calculus that deals with functions represented as the ratio of two polynomials. The form of a rational function is expressed as \( f(x) = \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials and \( Q(x) eq 0 \).

When analyzing rational functions, it's critical to determine their domain, which is the set of all possible values of \( x \) that make the function well-defined. A key part of this analysis is identifying the values of \( x \) that cause the denominator to equal zero since division by zero is undefined in mathematics. These values are excluded from the domain and can lead to vertical asymptotes, which we will explore in more detail in the following section on asymptotes of functions.

To analyze rational functions effectively, mathematicians often examine end behavior, investigate intercepts with the axes, and look for symmetry and asymptotes, which provide a comprehensive understanding of these functions' behavior.

Asymptotes of Functions

Asymptotes of functions are lines that the graph of a function approaches but never actually reaches. Vertical asymptotes occur at points on the x-axis where the function grows without bound as it approaches from either the left or the right. A common cause of vertical asymptotes is the division by zero within rational functions, although other types of functions can also have vertical asymptotes.

To find vertical asymptotes, we typically set the denominator equal to zero and solve for the variable. However, if the denominator can't be zero for any real number, as with the function \( g(t)=\frac{t-1}{t^{2}+1} \) where the denominator \( t^{2} + 1 \) is always positive, then the function has no vertical asymptotes.

It's worth noting that not all values that make the denominator zero will result in a vertical asymptote; these could also be points of removable discontinuity, depending on the behavior of the numerator. Therefore, it's important to closely examine the function's entire equation when conducting this analysis.