Rational Functions

Domain of Rational Functions

The domain of a rational function consists of all real numbers except those that make the denominator zero. To find the domain, set the denominator equal to zero and solve for the excluded values.

• Example: For , set and solve for .

• Excluded values:

• Domain:

Intercepts of Rational Functions

Intercepts are points where the graph crosses the axes.

• y-intercept: Set and solve for .

• x-intercept: Set and solve for .

• Example: For , x-intercept is found by setting .

Vertical, Horizontal, and Oblique Asymptotes

Vertical asymptotes occur at values of that make the denominator zero (and not canceled by the numerator). Horizontal asymptotes are determined by the degrees of the numerator and denominator. Oblique asymptotes occur when the degree of the numerator is one more than the denominator.

• Vertical asymptote: Set denominator and solve for .

• Horizontal asymptote: Compare degrees:

  • If degree numerator < degree denominator:

  • If degrees equal:

• Oblique asymptote: Use polynomial long division if degree numerator = degree denominator + 1.

Logarithmic and Exponential Equations

Solving Logarithmic Equations

To solve equations involving logarithms, use properties such as:

Example: Solve

• Combine:

• Rewrite:

• Solve:

Exponential Growth and Decay

Exponential models describe growth or decay processes. The general form is:

• Growth:

• Decay (half-life): , where is the half-life

Example: If the half-life of silicon-32 is 70 years and 30 grams is present now, how much will be present in 200 years?

• Calculate to get the remaining amount.

Trigonometric Functions and Equations

Solving Trigonometric Equations

Trigonometric equations can be solved using identities and algebraic manipulation. Solutions are often found within a specified interval, such as .

• Example:

• Solve:

• Find all in where

Trigonometric Identities

Identities are equations that are true for all values in the domain. Common identities include:

• Pythagorean:

• Double Angle:

• Sum and Difference:

Simplifying Trigonometric Expressions

Use identities to simplify expressions.

• Example:

• Rewrite and combine terms.

Graphing Functions

Graphing Rational and Exponential Functions

To graph a function, identify intercepts, asymptotes, and behavior at infinity. Plot key points and sketch the curve accordingly.

• Example: For , find vertical asymptotes at and , and horizontal asymptote at .

Tables

Sample Table: Asymptotes and Intercepts of Rational Functions

Additional info:

• Some questions involve matching graphs to rational functions based on intercepts and asymptotes.

• Exponential growth models use , where is the growth rate.

• Half-life problems use exponential decay formulas.

• Trigonometric equations may have multiple solutions within a given interval.