Question

Find the horizontal and vertical asymptotes of the graph of the function \(f(x)=\frac{1}{x^{2}+x-2} .\)

Short answer

Answer: The function has two vertical asymptotes at \(x = -2\) and \(x = 1\), and one horizontal asymptote at \(y = 0\).

Step-by-step solution

Identify the function

The given function is \(f(x) =\frac{1}{x^{2}+x-2}\).

Find vertical asymptotes

To find the vertical asymptotes, we need to find the values of \(x\) for which the denominator of the function is equal to zero. The denominator is \(x^2 + x - 2\). Solve the equation \(x^2 + x - 2 = 0\): Factor the quadratic equation: \((x+2)(x-1) = 0\) Thus, \(x=-2\) and \(x=1\) are the zeros of the denominator. These are the vertical asymptotes. So we have two vertical asymptotes at \(x = -2\) and \(x = 1\).

Find horizontal asymptotes

To find the horizontal asymptotes, we need to analyze the behaviour of the function as \(x\) approaches infinity (positive and negative) and check whether the function approaches a constant value. Analyze the function as \(x\) approaches positive infinity: \(\lim _{x\rightarrow +\infty} \frac{1}{x^{2}+x-2}\) Since the highest power of \(x\) in the denominator is 2, the fraction approaches zero as \(x\) goes to positive infinity: \(\lim_{x\rightarrow +\infty} \frac{1}{x^{2}+x-2} = 0\) Analyze the function as \(x\) approaches negative infinity: \(\lim _{x\rightarrow -\infty} \frac{1}{x^{2}+x-2}\) Similarly, the fraction also approaches zero as \(x\) goes to negative infinity: \(\lim_{x\rightarrow -\infty} \frac{1}{x^{2}+x-2} = 0\) Therefore, there is a horizontal asymptote at \(y = 0\).

Write the final answer

The given function has two vertical asymptotes at \(x = -2\) and \(x = 1\), and one horizontal asymptote at \(y = 0\).

Key concepts

Horizontal Asymptote

Horizontal asymptotes describe the behavior of a rational function as the input variable, often denoted as \(x\), goes to positive or negative infinity. They tell us where the output \(y\) will settle. For the function \(f(x) = \frac{1}{x^{2}+x-2}\), the horizontal asymptote occurs when the degree of the polynomial in the numerator is less than the degree of the polynomial in the denominator.

In this case, the numerator is 1 (degree 0), and the denominator is \(x^2 + x - 2\) (degree 2). This means as \(x\) becomes very large or very small, the function approaches \(y = 0\).

• When \(x\) approaches either positive or negative infinity, the function behaves similarly because of the squared term.
• This results in a horizontal asymptote at \(y = 0\).

Vertical Asymptote

Vertical asymptotes are points where a function tends toward infinity. They occur where the denominator of a rational function is zero and the numerator is non-zero. For \(f(x) = \frac{1}{x^{2}+x-2}\), we find the vertical asymptotes by factoring the denominator.

Solving \(x^2 + x - 2 = 0\) gives us \((x+2)(x-1) = 0\), leading to \(x = -2\) and \(x = 1\). Thus, the function has vertical asymptotes at these values:

• As \(x\) approaches \(-2\) or \(1\), the function value increases or decreases without bound, creating a vertical asymptote.
• This is because the denominator becomes zero at these points, causing division by zero.

Rational Functions

Rational functions are fractions where both the numerator and denominator are polynomials. They are crucial in calculus due to their asymptotic behavior. With the function \(f(x) = \frac{1}{x^{2}+x-2}\), we see a classic example of a rational function.

These functions can have both horizontal and vertical asymptotes.

• The numerator and denominator control the shape and asymptotes.
• In our example, the numerator is constant, but the denominator is a quadratic, highlighting its role in creating asymptotes.

Rational functions can appear complex, but understanding their asymptotes helps predict their behavior across different intervals.

Limits

The concept of limits helps us understand the behavior of functions as they approach specific points or infinity. In the context of asymptotes, limits allow us to determine the value a function approaches.

For \(f(x) = \frac{1}{x^{2}+x-2}\), we use limits to find both horizontal and vertical asymptotes.

• To find horizontal asymptotes, we calculate \(\lim_{x \to \infty} f(x)\) and \(\lim_{x \to -\infty} f(x)\), both yielding 0 in this case.
• Vertical asymptotes are identified at the points where limits of the function approach infinity, typically where the denominator equals zero.

Limits are a fundamental concept in calculus, aiding in the precise understanding of function behavior.

Quadratic Equations

Quadratic equations are polynomials of degree 2, represented by the general form \(ax^2 + bx + c = 0\). They often arise in the context of rational functions, dictating the behavior of the denominator.

In \(f(x) = \frac{1}{x^{2}+x-2}\), \(x^2 + x - 2\) is the quadratic equation. To find solutions, we factor it: \((x+2)(x-1) = 0\), giving roots where the function is undefined.

• This breakdown into factors reveals the vertical asymptotes.
• Solving quadratic equations involves methods like factoring, using the quadratic formula, or completing the square.

Understanding these equations is crucial for recognizing how they influence rational functions.