Functions and Their Properties

Evaluating Functions and Piecewise Functions

In College Algebra, understanding how to evaluate functions, including piecewise-defined functions, is essential. A function assigns each input value to exactly one output value, and piecewise functions use different rules for different intervals of the domain.

• Function Evaluation: To evaluate a function, substitute the given input value into the function's formula. For example, if , then .

• Piecewise Functions: These are defined by different expressions depending on the input value. For example:

• Application: Piecewise functions are often used to model situations where a rule changes at a certain point, such as tax brackets or shipping rates.

• Interpreting Functional Values: In word problems, functional values often represent quantities such as cost, distance, or population at a given time.

Graphical Analysis of Functions

Analyzing the graph of a function provides insight into its behavior, including domain, range, intercepts, and intervals of increase or decrease.

• Domain and Range: The domain is the set of all possible input values (x-values), and the range is the set of all possible output values (y-values).

• Intervals of Increase/Decrease: A function is increasing where its graph rises as x increases, and decreasing where it falls.

• Local Maximum/Minimum: A local maximum is a point where the function reaches a highest value in a neighborhood, and a local minimum is where it reaches a lowest value.

• x- and y-intercepts: The x-intercept is where the graph crosses the x-axis (), and the y-intercept is where it crosses the y-axis ().

• Interval Notation: Used to express domains and ranges, such as or .

Domain Restrictions and Interval Notation

Some functions have domain restrictions, such as division by zero or taking the square root of a negative number. Interval notation is used to clearly state these domains.

• Example: The domain of is all real numbers except , written as .

• Interval Notation: Parentheses indicate endpoints are not included; brackets indicate endpoints are included.

Average Rate of Change

The average rate of change of a function over an interval measures how much the output changes per unit change in input. It is analogous to the slope of a line connecting two points on the graph.

• Formula:

• Numerical Example: For , the average rate of change from to is:

• Generalized Interval: The interval can be expressed as to for a more abstract calculation.

Function Composition

Function composition involves combining two functions so that the output of one function becomes the input of another. This is denoted as .

• Notation: means apply to , then apply to the result.

• Example: If and , then .

• Evaluating Composition: You may be asked to evaluate compositions for specific values, such as .

• Applications: Composition is used in modeling multi-step processes, such as converting units and then applying a formula.

Table: Function Properties and Notation