Question

Identify the horizontal and vertical asymptotes, if any, of the given function. $$ f(x)=\frac{x^{2}-1}{-x^{2}-1} $$

Short answer

Horizontal asymptote at \(y = -1\); no vertical asymptotes.

Step-by-step solution

Understand Asymptotes

Asymptotes are lines that the graph of a function approaches but never touches. Horizontal asymptotes are related to the end behavior of the function, while vertical asymptotes occur where the function is undefined due to division by zero. We will analyze the given function to identify these points.

Simplify the Function

Simplify the function, if possible, to make it easier to identify asymptotes. In this case, the function is already simplified: \[ f(x) = \frac{x^2 - 1}{-x^2 - 1} \]

Find Vertical Asymptotes

To find vertical asymptotes, set the denominator equal to zero and solve for \(x\):\[-x^2 - 1 = 0 \]\[x^2 = -1 \]This equation has no real solutions, so there are no vertical asymptotes.

Find Horizontal Asymptotes

For horizontal asymptotes, compare the degrees of the numerator and the denominator. Both are degree 2, and the leading coefficients are 1 and -1, respectively. Thus, \[ y = \frac{1}{-1} = -1 \]So, there is a horizontal asymptote at \(y = -1\).

Key concepts

Horizontal Asymptotes

Horizontal asymptotes are crucial in understanding the long-term behavior of a function as the input variable, usually denoted as \(x\), goes to infinity or negative infinity. The concept is all about what happens to the values of a function as \(x\) grows very large in the positive or negative direction. Essentially, a horizontal asymptote is a horizontal line \(y = k\) that the graph of a function may approach, but never actually reaches, as \(x\) tends towards positive or negative infinity.
To determine if a horizontal asymptote exists, compare the degrees of the numerator and denominator polynomials in a rational function.

• If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \(y = 0\).
• If the degrees are equal, the horizontal asymptote will be the ratio of the leading coefficients of these polynomials.
• If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

Applying these rules to the function \(f(x) = \frac{x^2 - 1}{-x^2 - 1}\), notice that both the numerator and denominator are of degree 2. This means the horizontal asymptote is found by dividing the leading coefficients, which are 1 and -1, respectively. Therefore, the horizontal asymptote is \(y = -1\).

Vertical Asymptotes

Vertical asymptotes occur in rational functions when the function becomes undefined due to division by zero. These asymptotes appear as vertical lines \(x = a\) where the function can increase or decrease without bound, essentially making these points forbidden on the graph.
To find vertical asymptotes in a function like \(f(x) = \frac{x^2 - 1}{-x^2 - 1}\), set the denominator equal to zero and solve for \(x\). For this function, we deal with the equation \(-x^2 - 1 = 0\). Solving for \(x\) yields \(x^2 = -1\). Notice that this equation has no real solutions since a real number squared cannot equal a negative number. Therefore, \(f(x)\) has no vertical asymptotes.
It is important to note that the presence of complex solutions like \(x = \pm i\) does not affect the existence of vertical asymptotes defined by real numbers. The focus remains on where the function is undefined due to real division by zero.

End Behavior of Functions

Understanding the end behavior of functions helps in visualizing how a function behaves as \(x\) approaches extreme values, either positive or negative infinity. This behavior is largely dictated by the function's horizontal asymptotes if they exist.
For the function \(f(x) = \frac{x^2 - 1}{-x^2 - 1}\), as \(x\) becomes very large (either positively or negatively), the effect of lower-degree terms such as constant terms or linear terms diminishes. Thus, the function's value approaches the horizontal asymptote, \(y = -1\). This signifies that no matter how large or small \(x\) gets, the output of the function will come increasingly close to \(-1\), but never quite reach it.
End behavior can be portrayed by analyzing limits. For instance, \(\lim_{x \to \infty} f(x)\) and \(\lim_{x \to -\infty} f(x)\) provide insight into how the function behaves as it travels to the edges of its graph. In conclusion, the end behavior of \(f(x)\) denotes that the values settle towards the line \(y = -1\) at both ends. Ultimately, grasping end behavior is vital to understanding the overall nature and trajectory of a function's graph.