Question

Finding Vertical Asymptotes In Exercises \(13-28\) , find the vertical asymptotes (if any) of the graph 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

Identify the Denominator

The denominator of function \(g(t)\) is \(t^{2}+1\). The vertical asymptotes occur when the denominator equals zero and it's not simultaneously making the numerator zero.

Solve for t

Solve the equation \(t^{2}+1 = 0\) for \(t\). This results in \(t^{2}= -1\). After solving for \(t\), you will see that there are no real solutions because a square of a real number cannot be a negative number.

Check for Vertical Asymptotes

Since there are no real solutions, this means there are no vertical asymptotes for the function \(g(t)=\frac{t-1}{t^{2}+1}\).

Key concepts

Rational Functions

Rational functions are mathematical expressions that represent the ratio of two polynomials. They take the form of \( f(x) = \frac{P(x)}{Q(x)} \) where \( P(x) \) and \( Q(x) \) are polynomials with \( Q(x) \) not equal to zero. Understanding the characteristics of these functions is key in finding their asymptotes, which are lines that a graph approaches but never reaches.
For example, in the function \( g(t)=\frac{t-1}{t^{2}+1} \), the numerator is \( t-1 \), and the denominator is \( t^{2}+1 \). The behavior of the graph around the points where the denominator is zero is essential for finding vertical asymptotes. However, since a polynomial like \( t^{2}+1 \) never equals zero for real numbers, this specific function has no vertical asymptotes. Remember to always look for values that make the denominator equal to zero, except where it cancels with the numerator, to find vertical asymptotes of a rational function.

Graph Behavior

Understanding the behavior of a graph near its asymptotes is a fundamental aspect of analyzing rational functions. Graphs can approach these asymptotes either from above or below, and either horizontally or vertically. A vertical asymptote typically occurs at values of \( x \) that cause the function's denominator to be zero, as the function's value tends toward infinity.
However, not all functions with a denominator will have vertical asymptotes. For example, the function \( g(t)=\frac{t-1}{t^{2}+1} \) doesn't exhibit vertical asymptotes because the denominator never reaches zero in the real number system. Instead of searching for where the graph meets infinity, you would observe that the graph smoothly continues over the entire range of \( t \), crossing \( t=1 \) where the numerator equals zero. Grasping this behavior helps when sketching graphs and anticipating how they will look, even before plotting any points.

Complex Numbers

Complex numbers are an extension of the real numbers and can be useful when dealing with polynomials that do not have real solutions. A complex number has the form \( a + bi \) where \( a \) is the real part, \( b \) is the imaginary part, and \( i \) is the imaginary unit defined by the property that \( i^{2} = -1 \). When solving equations like \( t^{2} + 1 = 0 \), which appears in our example function \( g(t)=\frac{t-1}{t^{2}+1} \), we encounter the square root of a negative number.
In this case, no real number squared will give a negative result, indicating that our solutions lie in the realm of complex numbers. Specifically, the solutions are \( t = i \) and \( t = -i \), which are not on the real number line. This is why there are no vertical asymptotes in the real graph of \( g(t) \), although if complex numbers were graphed, these would be the points of interest. Understanding complex numbers is pivotal for working with functions that have roots not found on the standard number line.