Function Review

Domain and Range

Understanding the domain and range of a function is fundamental in precalculus. The domain is the set of all possible input values (x-values) for which the function is defined, while the range is the set of all possible output values (y-values) the function can produce.

• Domain: All x-values for which the function produces a real output.

• Range: All y-values that the function can take as x varies over the domain.

• Example: For , the domain is and the range is .

Expanding and FOIL

Expanding expressions, especially binomials, is a key algebraic skill. The FOIL method is used to expand the product of two binomials.

• FOIL: First, Outer, Inner, Last terms are multiplied and then summed.

• Example:

Function Notation

If , then y is the output value when x is substituted into the function f.

• Example: If , then .

Parabolas

Intercepts and Vertex

A parabola is the graph of a quadratic function, typically written as .

• x-intercepts: Points where the graph crosses the x-axis (). Solve .

• y-intercept: Point where the graph crosses the y-axis (). Compute .

• Vertex: The maximum or minimum point of the parabola. The x-coordinate is .

• Plug the x-coordinate into the function to find the y-coordinate of the vertex.

• Example: For , vertex at , .

Graphing Parabolas

• Use the vertex, intercepts, and symmetry to sketch the graph.

• If given the x-intercepts, y-intercept, and vertex, you can reconstruct the equation of the parabola.

• Domain: All real numbers ().

• Range: If , range is ; if , range is , where is the y-coordinate of the vertex.

Rational Functions and Their Graphs

Definition and Simplification

A rational function is a function of the form , where P and Q are polynomials and .

• Simplifying: Factor numerator and denominator, then cancel common factors.

• Domain: All real numbers except those that make the denominator zero.

• Example: , ; simplifies to for .

Finding the Domain

• Set the denominator equal to zero and solve for x. Exclude these values from the domain.

• Example: ; , .

Graphing Rational Functions

• Identify vertical asymptotes (where denominator is zero and not canceled by numerator).

• Identify holes (where a factor cancels in numerator and denominator).

• Find horizontal asymptotes by comparing degrees of numerator and denominator.

• Sketch the graph, marking intercepts, asymptotes, and holes.

Asymptotes and Holes

• Vertical Asymptote: At if and .

• Horizontal Asymptote: Determined by degrees of numerator (n) and denominator (m):

  • If , asymptote at .

  • If , asymptote at .

  • If , no horizontal asymptote (may have an oblique/slant asymptote).

• Hole: At if is a common factor in numerator and denominator.

Polynomial Functions and Roots

Roots, Zeros, and Imaginary Numbers

The roots (or zeros) of a polynomial are the values of x that make the polynomial equal to zero.

• Root/Zero:

• Real roots: Correspond to x-intercepts on the graph.

• Imaginary roots: If is a root, then is also a root (complex roots occur in conjugate pairs).

Key Theorems and Tools

• Descartes' Rule of Signs: Predicts the number of positive and negative real roots based on sign changes in the coefficients.

• Rational Root Theorem: Possible rational roots of are .

• Theorem on Bounds: Provides bounds within which all real roots must lie.

• Synthetic Division: A shortcut for dividing polynomials by linear factors of the form .

Constructing Polynomials

• Given roots, intercepts, or other information, you can reconstruct the equation of a polynomial.

• Example: If a cubic has roots at , then for some constant .

Calculator and Test Preparation Guidelines

• Graphing calculators are expected for class and exams.

• One note sheet (one side, full-sized) is allowed during tests; only formulas and written concepts, no homework solutions.

• Practice both with calculators and by hand to prepare for timed tests.

• Tests are 50 minutes; be prepared to work efficiently.

• Follow all exam rules regarding electronics and materials.

Study and Practice Recommendations

• Attend class and complete all assignments on time.

• Take notes on homework problems and topics needing review.

• Complete review assignments and practice problems, especially those involving multiple concepts.

Additional info: Some theorems and methods (Descartes' Rule, Rational Root Theorem, Theorem on Bounds, Synthetic Division) are referenced but not fully explained in the original notes; brief academic context has been added for completeness.