Algebra lecture 3/6

Introduction to Rational Functions

• Definition: Rational function is the quotient of two polynomial functions, expressed as f(x) = p(x) / q(x).

Domain of Rational Functions

• Identifying Exceptions: Evaluate the function to find "exceptions" where the function is undefined.

  • Example: Solving p(x) = 0 gives exceptions (x = 5 in this case).

  • Domain: All real numbers except x = 5.

  • In interval notation: (-∞, 5) ∪ (5, ∞).

• Case with No Exceptions: If the equation has no real solutions (e.g. x² + 1 = 0), the domain is (-∞, ∞).

Asymptotes

• Vertical Asymptotes: Occur at x = a where f(x) is undefined. To find, set q(x) = 0 after ensuring p(x) and q(x) are factored without common factors.

  • Example: For f(x) = x / (x² - 1), vertical asymptotes occur where the denominator is zero.

• Horizontal Asymptotes: Determined by comparing the degrees of the polynomials in p(x) and q(x).

  • If n (degree of numerator) > m (degree of denominator), no horizontal asymptote.

  • If n = m, asymptote is y = (leading coefficient of p) / (leading coefficient of q).

  • Examples:

    • For f(x) = 9x² / (3x² + 1), degree is equal; thus, y = 9/3 = 3.

Arrow Notation

• Arrow Notation: Represents the limiting behavior of functions.

  • e.g. "x approaches a" indicates where x is getting close to a certain value.

  • Notations:

    • x → ∞: x increases without bound.

    • x → -∞: x decreases without bound.

    • x → a⁻: x approaches a from the left.

    • x → a⁺: x approaches a from the right.

Graphing Rational Functions

• Transformations: Understand that the graph's transformations impact both the vertical and horizontal asymptotes.

  • Example: x + 2 shifts the graph left by 2.

• Symmetry: Determine if the function is even, odd, or neither based on its structure:

  • Odd functions have origin symmetry (e.g., f(x) = 1/x).

  • Even functions have y-axis symmetry (e.g., f(x) = 1/x²).

Finding Intercepts and Points

• Y-Intercept: Evaluated by f(0).

• X-Intercepts: Set the numerator equal to zero (solve for x).

  • Use points between x-intercepts and beyond to plot the graph thoroughly.

Test-taking Strategies for Rational Functions

• Recommendation: Solve for simpler problems (horizontal asymptotes) first to gain confidence and partial credit.

Conclusion

• Understanding rational functions includes identifying vertical & horizontal asymptotes, effective use of domain, transformations for graphing, and utilizing arrow notation for limits in behavior.