In Exercises 57–80, follow the seven steps to graph each rational function. f(x)=2/(x²+x−2)

Verified step by step guidance

1. Identify the domain of the function by finding the values of $x$ that make the denominator zero. Solve the equation $x^2+x-2=\; 0$ to find these values.
2. Factor the denominator $x^2+x-2$ into two binomials to simplify the expression and identify vertical asymptotes.
3. Determine the vertical asymptotes by setting each factor of the denominator equal to zero and solving for $x$. These are the values where the function is undefined.
4. Find the horizontal asymptote by comparing the degrees of the numerator and denominator. Since the numerator is a constant and the denominator is quadratic, the horizontal asymptote is $y\; =\; 0$.
5. Choose test points in each interval determined by the vertical asymptotes to evaluate $f(x)$ and understand the behavior of the graph near the asymptotes and across the domain.

Key Concepts

Rational Functions

A rational function is a ratio of two polynomials, expressed as f(x) = P(x)/Q(x). Understanding its domain, where the denominator Q(x) ≠ 0, is essential because values that make the denominator zero are excluded and often correspond to vertical asymptotes or holes in the graph.

Factoring Polynomials

Factoring the denominator polynomial, such as x² + x − 2, helps identify zeros of the denominator. These zeros determine vertical asymptotes or holes in the graph, which are critical for sketching the function accurately and understanding its behavior near undefined points.

Asymptotes and Graph Behavior

Asymptotes are lines that the graph approaches but never touches. Vertical asymptotes occur where the denominator is zero, and horizontal or oblique asymptotes describe end behavior as x approaches infinity or negative infinity. Recognizing these helps in accurately graphing rational functions.