Functions and Their Properties

Limits and Function Values

Understanding limits and function values is essential for analyzing the behavior of functions, especially at points of discontinuity or endpoints.

• Limit of a function as x approaches a value describes the behavior of the function near that point, not necessarily the value at the point.

• Function value at a point is the actual output of the function for that input.

• Discontinuities occur when the limit does not equal the function value or the function is not defined at that point.

Example:

• For a piecewise graph, may differ from if there is a jump or hole at .

Polynomial Functions

Types of Polynomial Functions

Polynomials are classified based on their degree and structure.

• Linear Polynomial: Degree 1, e.g.,

• Non-linear Polynomial: Degree 2 or higher, e.g.,

• Rational Function: Ratio of two polynomials, e.g.,

Zeros of Polynomial Functions

Zeros (roots) are values of where .

• To find zeros, set the polynomial equal to zero and solve for .

• For , factor or use synthetic division to find -intercepts.

Example:

• has zeros at .

Vertex of a Quadratic Function

The vertex of a parabola defined by is the point where the function reaches its maximum or minimum.

• Vertex formula:

• Substitute into to find the -coordinate.

Example:

• For , vertex at .

End Behavior of Polynomials

The end behavior describes how the function behaves as or .

• Determined by the leading coefficient and degree.

• Leading Coefficient Test:

  • If degree is even and leading coefficient is positive, as .

  • If degree is odd and leading coefficient is positive, as , as .

Example:

• falls to the left and rises to the right.

Concavity and Inflection Points

Concave Upward and Downward

Concavity describes the direction of curvature of a function's graph.

• Concave upward: Graph opens upwards, .

• Concave downward: Graph opens downwards, .

Inflection Points

Inflection points are where the graph changes concavity.

• Occurs where and the sign of changes.

Example:

• From a graph, inflection points can be estimated visually where the curvature changes.

Rational Functions

Vertical and Horizontal Asymptotes

Asymptotes are lines that the graph approaches but never touches.

• Vertical asymptote: Occurs at where the denominator is zero and the numerator is not zero.

• Horizontal asymptote: Determined by the degrees of numerator and denominator.

  • If degrees are equal, .

  • If degree of numerator < denominator, .

  • If degree of numerator > denominator, no horizontal asymptote (may be slant/oblique).

Example:

• has horizontal asymptote .

Oblique (Slant) Asymptotes

Occurs when the degree of the numerator is exactly one more than the degree of the denominator.

• Found by polynomial long division.

Example:

• has a slant asymptote.

Finding Asymptotes and Holes

• Vertical asymptotes: Set denominator equal to zero and solve for .

• Holes: Occur when a factor cancels in numerator and denominator.

Example:

• has a hole at and .

Graph Analysis

Multiplicity of Zeros

Multiplicity refers to the number of times a zero occurs.

• If a zero has even multiplicity, the graph touches and turns around at the -axis.

• If a zero has odd multiplicity, the graph crosses the -axis.

Critical Points in Inequalities

Critical points are values where the function changes sign, important for solving inequalities.

• Set each factor equal to zero to find critical points.

Example:

• For , critical points are and .

Optimization Problems

Maximizing Area with Constraints

Optimization involves finding the maximum or minimum value of a function given certain constraints.

• Express the quantity to be maximized/minimized as a function of one variable.

• Use calculus or algebraic methods to find the maximum/minimum.

Example:

• Maximize area of a rectangle with fixed perimeter using and constraint .

Polynomial Long Division

Dividing Polynomials

Long division is used to divide polynomials, especially for finding slant asymptotes or simplifying rational expressions.

• Divide the highest degree term of the numerator by the highest degree term of the denominator.

• Multiply, subtract, and repeat until the remainder is of lower degree than the divisor.

Example:

• Divide by .

Summary Table: Asymptotes and Holes in Rational Functions

Additional info:

• Some questions involve graphical analysis, requiring students to interpret graphs for limits, function values, and asymptotes.

• Optimization and critical point problems are foundational for calculus and further mathematical study.