Precalculus Functions and Their Representations

Precalculus Function Concepts

Definitions

  • Relation: A set of ordered pairs, which can represent any relationship between two sets.
  • Function: A special type of relation where each input (domain) is associated with exactly one output (range).
  • Domain: The complete set of possible values of the independent variable.
  • Range: The complete set of possible values of the dependent variable obtained from the function.
  • Independent Variable: The variable that represents the input values; typically denoted as x.
  • Dependent Variable: The variable that represents the output values; typically denoted as y.

Ways to Represent a Function

  1. Graphically: Using graphs to plot ordered pairs.
  2. Algebraically: Using equations or expressions to define relationships between variables.
  3. Numerically: Through tables of values that correspond to input-output pairs.
  4. Verbal: Describing the function in words to provide context and understanding.

Examples of Functions

Example 1: Determining Functionality

  • a. The input value x represents the number of representatives from a state, and the output variable y represents the number of senators from a state.
  • Assess whether this relationship defines y as a function of x.

Example 2: Algebraic Representation of Function

  • The equation y = x^2 represents the variable y as a function of the variable x.
  • Here, x is the independent variable and y is the dependent variable.
  • The domain is determined by the values that x can take, typically all real numbers unless restrictions are applied.
  • The range is determined by the values that y can take, in this case y ext{ is } ext{ non-negative}.

Example 3: Assessing Function Representation

  • Determine if these equations express y as a function of x:
    • a. x^2 + y = 1: Not a function.
    • b. -x + y^2 = 1: Not a function.

Function Notation

  • Functions are commonly expressed with function notation, denoted as f(x).
    • Input: Value assigned to x.
    • Output: Result after applying the function to x.

Example of Function Evaluation

  • Let g(x) = -x^2 + 4x - 1.
  • Find:
    • a. g(2)
    • Evaluation: g(2) = -2^2 + 4(2) - 1 = -4 + 8 - 1 = 3.
    • b. g(t) resulting in the expression g(t) = -t^2 + 4t - 1.
    • c. g(x + 2) results in g(x + 2) = - (x + 2)^2 + 4(x + 2) - 1 which needs expansion.
    • d. Evaluate rac{g(x + h) - g(x)}{h}.

Piecewise Function Definition

  • A function defined by two or more equations over a specified domain is called a piecewise function.
  • Example: Evaluate the function when x = -1, 0, 1 for the piecewise function:
    f(x) = \begin{cases} x^2 + 1 & \text{if } x < 0 \ x - 1 & \text{if } x \geq 0 \end{cases}

Finding Function Values

Example 4: Finding Zeros of Functions

  • Find all real values of x such that:
    • a. f(x) = -2x + 10 = 0
    • Solve: -2x + 10 = 0
      ightarrow x = 5.
    • b. f(x) = x^2 - 5x + 6 = 0
    • Factor: (x - 2)(x - 3) = 0
      ightarrow x = 2, 3.

Example 5: Finding Intersections of Functions

  • Find all values for which f(x) = g(x):
    • a. f(x) = x^2 + 1 and g(x) = 3x - x^2
    • Set equal: x^2 + 1 = 3x - x^2 leads to 2x^2 - 3x + 1 = 0.
    • b. f(x) = x^2 - 1 and g(x) = -x^2 + x + 2
    • Set equal: x^2 - 1 = -x^2 + x + 2 leads to solved quadratic.

Domain and Range

Determining Domain

  • Domains can be explicitly or implicitly determined:
    • Explicitly: Specifying the interval or set of values.
    • Implicitly: Deducing from the context of the function or relation provided.
  • Domain = Specific intervals or exclusions determined.

Example 6: Domain and Range Assessment

  • Find the domain and range for:
    • 1. Defined by explicit equations.
    • 2. Set of ordered pairs: {(-3, 1), (2, -4), (0, 5), (-1, 1), (3, 4)}
    • 3. Function: g(x) = rac{1}{x - 5} which is defined for all x except x = 5.
    • 4. Volume equation: V = rac{4}{3} imes ext{π} r^3 where r should be non-negative.

Application Example

Example of Projectile Motion

  • Scenario: A baseball is hit 3 feet above the ground with a velocity of 100 feet per second at an angle of 45 degrees.
  • Path: Given by the function:
    f(x) = -0.0032x^2 + x + 3 where both x (horizontal distance) and f(x) (height) are in feet.
  • Query: Will the baseball clear a 10-foot high fence that is located 300 feet from home plate?
    • Calculate f(300) and check if f(300) > 10.
    • Comparison: If true, it will clear the fence; otherwise, it will not.