GED Math - S04.1V3
Functions > Linear Functions
Important Instructions
"Before starting the test, let's review the essential concepts of Functions. Please read each question carefully and respond to all 22 questions within TIME LIMIT of 25 min.
Understanding Functions: Identifying Functions, Function Notation, and Graph Interpretation
1. Identifying Functions
Definition: A function is a relation where each input (x-value) has exactly one output (y-value).
- Vertical Line Test: A graph represents a function if no vertical line intersects the graph at more than one point.
Examples:
- The set of ordered pairs \( \{(1, 2), (2, 4), (3, 6)\} \) is a function because each input has one output.
- The relation \( \{(1, 2), (1, 3)\} \) is not a function because the input \( x = 1 \) corresponds to multiple outputs.
2. Understanding Function Notation
Definition: Function notation uses \( f(x) \) to denote a function where \( x \) is the input, and \( f(x) \) is the corresponding output.
- Example: \( f(x) = 2x + 3 \)
- Interpretation: To find \( f(2) \), substitute \( x = 2 \):
\( f(2) = 2(2) + 3 = 7 \)
Key Points:
- Function notation specifies the rule for determining outputs from inputs.
- The variable in \( f(x) \) is a placeholder; \( f(a) \), \( f(t) \), or \( f(\text{value}) \) all represent the same function rule with different inputs.
3. Interpreting the Graph of a Function
Key Features:
- X-Intercepts: Points where the graph crosses the x-axis (\( f(x) = 0 \)).
- Y-Intercept: The point where the graph crosses the y-axis (\( x = 0 \)).
- Domain: All possible x-values of the function.
- Range: All possible y-values of the function.
Example:
- The graph of \( f(x) = x^2 - 4 \) has:
- X-Intercepts: \( (-2, 0) \) and \( (2, 0) \)
- Y-Intercept: \( (0, -4) \)
- Domain: \( (-\infty, \infty) \)
- Range: \( [0, \infty) \)
Final Tip
Mastery Tip: A strong understanding of functions builds a foundation for algebra, calculus, and real-world problem-solving. Always check for consistency in inputs and outputs, and use graphs to visualize relationships.
