Rational Functions

Definition and Basic Properties

A rational function is any function that can be written as the ratio of two polynomials, that is, , where . A polynomial is a special case of a rational function where the denominator is 1.

• Example: is a basic rational function.

• Exponent Rule: For any integer , .

Key Features of Rational Functions

• Domain: All real numbers except where the denominator is zero. At where , the function is undefined. This can result in a vertical asymptote or a hole in the graph.

• Vertical Asymptotes: Occur at values of that make the denominator zero, provided the factor does not cancel with the numerator.

• Holes: If a factor cancels from both numerator and denominator, a hole occurs at that value.

• Intercepts:

  • x-intercepts: Set the numerator equal to zero and solve for (provided the denominator is not zero at those points).

  • y-intercept: Evaluate , if defined.

• Horizontal Asymptotes: Determined by comparing the degrees of the numerator and denominator polynomials:

  • If degree numerator < degree denominator: is the horizontal asymptote.

  • If degrees are equal: .

  • If degree numerator > degree denominator: No horizontal asymptote (may have an oblique/slant asymptote).

Example: simplifies to for , but has a hole at .

Other Basic Algebraic Functions and Piecewise-Defined Functions

Root Functions

• Square Root Function:

  • Domain:

  • Range:

• Cubic Root Function:

  • Domain:

  • Range:

Absolute Value Function

• Definition:

  • Graph: V-shaped, vertex at (0,0)

  • Domain:

  • Range:

Piecewise-Defined Functions

• Definition: A function defined by different expressions over different intervals of the domain.

• Domain: Determined by the union of the intervals for which each piece is defined.

• Evaluation: To find for a given , use the formula corresponding to the interval containing .

• Graph Features: May have jumps or holes at the boundaries between pieces.

Example:

Composite Functions and Inverse Pairs

Composite Functions

• Definition: The composition means applying first, then to the result.

• Finding Formulas: Substitute into wherever appears.

• Decomposition: Given , identify and .

Example: If and , then .

Inverse Functions

• Definition: Two functions and are inverses if for all in their domains.

• Verification: Show both compositions simplify to .

• Existence: If a function is continuous and strictly increasing or decreasing, it has an inverse.

• Graphical Relationship: If is on the graph of , then is on the graph of .

• Exponential and Logarithmic Functions: These are inverse pairs for the same base.

Example: and are inverses.

Exponential Functions

Definition and Properties

• General Form: , where and .

• Domain:

• Range:

• Key Points: Passes through , , and .

• x-intercepts: None.

• y-intercept: Always at .

• Horizontal Asymptote: (for , on the left; for , on the right).

• Monotonicity: Increasing if , decreasing if .

• Continuity: Exponential functions are continuous everywhere.

• Exponent Rules:

Example:

The Natural Base

• Definition: (an irrational number).

• Natural Exponential Function:

• Importance: Widely used in calculus and mathematical modeling.

Transformations of Exponential Functions

• Shifts, stretches, and reflections can be applied to to obtain related graphs.

Solving Exponential Equations (Without Logarithms)

• Isolate the exponential expression and, if possible, express both sides with the same base to solve for the variable.

Example: Solve . Since , .

Introduction to Logarithmic Functions

Definition and Properties

• Definition: For , , if and only if .

• Natural Logarithm: if and only if .

• Common Logarithm: if and only if .

• Calculator Keys: Most calculators have keys for (base 10) and (base ).

Relationship Between Exponential and Logarithmic Functions

• Every logarithmic equation can be rewritten as an exponential equation and vice versa.

• Conversion:

  • From logarithmic to exponential:

  • From exponential to logarithmic:

Example:

Additional info: This guide expands on the original review outline by providing definitions, properties, and examples for each function type, as well as a summary table for comparison.