Functions and Their Properties

Evaluating Functions and Piecewise Functions

Understanding how to evaluate functions is a fundamental skill in College Algebra. This includes working with algebraically defined functions and piecewise-defined functions.

• Function Evaluation: To evaluate a function, substitute the given input value into the function's formula. For example, if f(x) = 2x + 3, then f(4) = 2(4) + 3 = 11.

• Piecewise-Defined Functions: These are functions defined by different expressions over different intervals of the domain. To evaluate, determine which interval the input belongs to and use the corresponding formula.

• Application: Functional values can be interpreted in the context of real-world problems, such as determining cost, distance, or other quantities based on input values.

• Example: If f(x) = x^2 for x < 0 and f(x) = 2x + 1 for x ≥ 0, then f(-2) = (-2)^2 = 4 and f(3) = 2(3) + 1 = 7.

Domain and Range of Functions

The domain of a function is the set of all possible input values (x-values), while the range is the set of all possible output values (y-values).

• Interval Notation: Used to express domains and ranges. For example, the domain 0 ≤ x ≤ 5 is written as [0, 5].

• Domain Restrictions: Some functions have restrictions, such as division by zero or taking the square root of a negative number.

• Example: The domain of f(x) = 1/(x-2) is all real numbers except x = 2, written as (-∞, 2) ∪ (2, ∞).

Graphical Analysis: Intercepts, Maxima, Minima, and Intervals

Analyzing the graph of a function helps identify key features such as intercepts, intervals of increase/decrease, and local extrema.

• x-intercept: The point(s) where the graph crosses the x-axis (f(x) = 0).

• y-intercept: The point where the graph crosses the y-axis (x = 0).

• Intervals of Increase/Decrease: Intervals where the function values are rising or falling as x increases.

• Local Maximum/Minimum: The highest or lowest point in a particular region of the graph.

• Example: For f(x) = -x^2 + 4, the vertex at (0, 4) is a local maximum.

Average Rate of Change

The average rate of change of a function over an interval measures how much the function's output changes per unit change in input.

• Formula:

• Application: This concept is similar to finding the slope of the secant line between two points on the graph.

• Example: For f(x) = x^2 from x = 1 to x = 5:

  • f(5) = 25, f(1) = 1

  • Average rate of change = (25 - 1)/(5 - 1) = 24/4 = 6

Function Composition

Composition of functions involves applying one function to the results of another. The notation (f ∘ g)(x) means f(g(x)).

• Notation: f(g(x)) or (f ∘ g)(x)

• Evaluating Composition: Substitute the output of g(x) into f(x).

• Example: If f(x) = 2x + 1 and g(x) = x^2, then f(g(3)) = f(9) = 2(9) + 1 = 19.

• Numerical Evaluation: You may be asked to compute values like g(f(2)) by first finding f(2) and then applying g to that result.

Additional info: Understanding the order of composition and the correct use of notation is essential for solving problems involving multiple functions.