Algebraic Functions and Equations

Polynomial Functions

Polynomial functions are a fundamental class of functions in precalculus, characterized by their algebraic expressions and predictable behavior. Understanding their properties is essential for analyzing graphs and solving equations.

• Definition: A polynomial function is an expression of the form , where and is a non-negative integer.

• Degree: The degree of a polynomial is the highest power of present.

• Roots/Zeros: The solutions to are called roots or zeros. The number of real roots depends on the degree and the nature of the coefficients.

• End Behavior: The end behavior of a polynomial function is determined by its leading term .

• Graphing: The graph of a polynomial is continuous and smooth, with turning points determined by the degree.

• Example: is a cubic polynomial.

Rational Functions

Rational functions are quotients of polynomials and exhibit unique properties such as asymptotes and discontinuities.

• Definition: A rational function is of the form , where and are polynomials and .

• Domain: The domain excludes values where .

• Asymptotes: Vertical asymptotes occur at zeros of ; horizontal or oblique asymptotes depend on the degrees of and $Q(x)$.

• Example: has a vertical asymptote at .

Exponential and Logarithmic Functions

Exponential Functions

Exponential functions model rapid growth or decay and are defined by a constant base raised to a variable exponent.

• Definition: An exponential function is , where and .

• Properties:

  • Domain: All real numbers

  • Range: Positive real numbers

  • Growth/Decay: If , the function grows; if , it decays.

• Example: is an exponential growth function.

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions and are used to solve equations involving exponents.

• Definition: is the logarithm base of , where , , and .

• Properties:

  • Domain:

  • Range: All real numbers

  • Inverse:

• Example: is the common logarithm.

Properties of Logarithms

Logarithms follow specific algebraic properties that simplify expressions and solve equations.

• Product Rule:

• Quotient Rule:

• Power Rule:

• Change of Base:

Solving Exponential and Logarithmic Equations

Solving these equations often requires applying properties of exponents and logarithms, and sometimes involves restrictions on the domain.

• Exponential Equations:

  • Set equal bases and equate exponents:

  • Take logarithms of both sides if bases differ.

• Logarithmic Equations:

  • Apply logarithm properties to combine or separate terms.

  • Check for extraneous solutions due to domain restrictions.

• Example: Solve . Since , .

Graphical Representations

Graphs of Polynomial, Rational, Exponential, and Logarithmic Functions

Graphing these functions helps visualize their behavior, including intercepts, asymptotes, and end behavior.

• Polynomial Graphs: Continuous, with turning points and end behavior determined by degree and leading coefficient.

• Rational Graphs: May have discontinuities and asymptotes.

• Exponential Graphs: Rapid increase or decrease, always positive.

• Logarithmic Graphs: Pass through , vertical asymptote at .

Summary Table: Properties of Functions

The following table summarizes key properties of polynomial, rational, exponential, and logarithmic functions.

Additional info:

Some content was inferred and expanded for completeness, including definitions, examples, and a summary table. The images included are directly relevant as they visually reinforce the review notes for algebraic, exponential, and logarithmic functions.