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Basic Functions

Definition

A function is a mathematical rule that assigns exactly one output value for each input value. It describes a relationship between an input set (domain) and an output set (range).

We often use notation like f(x) (read "f of x") to represent the output of a function f when the input is x.

Prerequisites

To understand basic functions, you should be familiar with:

  1. Variables & Expressions
  2. Basic Equation Solving
  3. Coordinate Plane
  4. Order of Operations

Learning Objectives

After mastering this topic, you should be able to:

  1. Define function, domain, and range.
  2. Identify whether a given relation (set of ordered pairs, graph, equation) represents a function (using the Vertical Line Test for graphs).
  3. Evaluate functions for specific input values (e.g., find f(3) given f(x) = 2x + 1).
  4. Determine the domain and range of simple functions (linear, basic quadratic).
  5. Interpret function notation in context.
  6. Graph basic linear functions.

Key Concepts

  • Function: A relation where each input (x-value) corresponds to exactly one output (y-value or f(x)-value).
  • Domain: The set of all possible input values (x-values) for which the function is defined.
  • Range: The set of all possible output values (y-values or f(x)-values) produced by the function.
  • Input: The value substituted into the function (often denoted by x).
  • Output: The value produced by the function (often denoted by y or f(x)).
  • Vertical Line Test: A visual test for graphs. If any vertical line intersects the graph more than once, the graph does not represent a function.
  • Function Notation: f(x) represents the output of function f for input x. For example, if f(x) = x + 5, then f(2) = 2 + 5 = 7.

Examples

Example 1: Evaluating a Function (Middle School)

Problem: If g(x) = 3x - 4, find g(5).

Solution: Replace x with 5 in the expression: g(5) = 3(5) - 4 = 15 - 4 = 11.

Example 2: Identifying Functions (Table) (Middle School)

Problem: Does the following set of ordered pairs represent a function? {(1, 2), (2, 4), (3, 6), (2, 5)}

Solution: No. The input value 2 corresponds to two different output values (4 and 5). This violates the definition of a function.

Example 3: Vertical Line Test (High School)

Problem: Use the Vertical Line Test to determine if the graph of a circle represents a function.

Solution: A circle fails the Vertical Line Test because a vertical line can intersect the circle at two points. Therefore, a circle does not represent y as a function of x.

Example 4: Domain and Range (Linear Function) (High School)

Problem: What are the domain and range of the function f(x) = -2x + 7?

Solution: * Domain: Since you can plug any real number into x and get a valid output, the domain is all real numbers (-∞, ∞). * Range: Since the output can be any real number (the line extends infinitely up and down), the range is all real numbers (-∞, ∞).

Common Misconceptions

  1. Confusing f(x) with f multiplied by x: f(x) is notation representing the output value, not multiplication.
  2. Assuming all equations are functions: Equations like x² + y² = 9 (a circle) or x = 3 (a vertical line) do not represent y as a function of x.
  3. Incorrectly identifying Domain/Range: Especially with functions that have restrictions (like square roots or denominators).

Applications in SAT

Functions are a major topic on the SAT, particularly in the "Passport to Advanced Math" section:

  1. Evaluating functions: Given f(x) and an input, find the output.
  2. Interpreting function notation: Understanding f(a) = b means when the input is a, the output is b.
  3. Analyzing graphs of functions: Identifying domain, range, intercepts, minimum/maximum values.
  4. Linear and Quadratic functions: Understanding their properties and graphs.
  5. Function composition: Understanding expressions like f(g(x)).

Advanced Connections

Basic functions are the foundation for:

  1. Advanced Functions - Exploring quadratic, polynomial, exponential, logarithmic, and rational functions.
  2. Trigonometry - Trigonometric functions (sine, cosine, tangent).
  3. Calculus: Analyzing rates of change (derivatives) and accumulation (integrals) of functions.
  4. Data Modeling: Using functions to represent real-world relationships.

Practice Problems

  1. Basic: If h(t) = 10 - 2t, find h(3).
  2. Intermediate: Does the graph of y = x² represent a function? Explain using the Vertical Line Test.
  3. Advanced: Find the domain of the function f(x) = √(x - 4). (Hint: The value inside a square root cannot be negative).
  4. SAT-Level: A function f is defined by f(x) = (x - 2)² + 5. For what value of x does f(x) reach its minimum value?