- Determine the set of all possible input values for a function based on its equation
- Identify the set of all possible inputs (domain) and outputs (range) from looking at a graph
- Figure out the allowed inputs and outputs for the fundamental toolkit functions
- Sketch piecewise functions, showing each segment with its own rule on the graph
Domain and Range
The Main Idea
- Domain:
- Set of all possible input values (x-values)
- Often represented using interval notation
- Range:
- Set of all possible output values (y-values)
- Determined by the function’s behavior
- Interval Notation:
- Uses brackets [ ] for inclusive endpoints
- Uses parentheses ( ) for exclusive endpoints
- Example: [latex](0, 100][/latex] means more than 0 and less than or equal to 100
- Domain Restrictions:
- Denominators: Exclude values making denominator zero
- Even roots: Exclude values making radicand negative
- Consider function’s context (e.g., real-world limitations)
- Range Analysis:
- Examine function behavior
- Consider limitations on output values
Key Techniques
- Finding Domain:
- For simple functions: Consider all real numbers
- For fractions: Set denominator to zero, solve for x, exclude those values
- For even roots: Set radicand ≥ 0, solve for x
- Finding Range:
- Analyze function behavior
- Consider minimum/maximum possible outputs
- Use graphing or tables to confirm
- Special Cases:
- Polynomials: Usually all real numbers for domain and range
- Rational functions: Exclude zeros in denominator for domain
- Root functions: Ensure non-negative radicand for domain
You can view the transcript for “Ex: Domain and Range of Square Root Functions” here (opens in new window).
Determine Domain and Range from a Graph
The Main Idea
- Visual Interpretation:
- Domain: All input values (x-axis)
- Range: All output values (y-axis)
- Graph Extent:
- Horizontal extent determines domain
- Vertical extent determines range
- Interval Notation:
- Write domain and range from left to right
- Use appropriate brackets/parentheses
- Graph Limitations:
- Consider unseen portions of the graph
- Be aware of potential continuations beyond visible area
Key Techniques
- Analyzing Continuous Graphs:
- Identify leftmost and rightmost x-values for domain
- Identify lowest and highest y-values for range
- Analyzing Discrete Graphs:
- Consider individual points for domain and range
- Pay attention to gaps or jumps in the graph
- Interpreting Asymptotes:
- Horizontal asymptotes affect range
- Vertical asymptotes affect domain
- Reading Scales:
- Note axis labels and units
- Estimate values between gridlines when necessary
You can view the transcript for “Ex 1 – Determine the Domain and Range of the Graph of a Function” here (opens in new window).
Domain and Range of Toolkit Functions
The Main Idea
- Constant Function: [latex]f(x) = c[/latex]
- Domain: All real numbers
- Range: [latex]{c}[/latex] or [latex][c, c][/latex]
- Identity Function: [latex]f(x) = x[/latex]
- Domain: All real numbers
- Range: All real numbers
- Absolute Value Function: [latex]f(x) = |x|[/latex]
- Domain: All real numbers
- Range: [latex][0, \infty)[/latex]
- Quadratic Function: [latex]f(x) = x^2[/latex]
- Domain: All real numbers
- Range: [latex][0, \infty)[/latex]
- Cubic Function: [latex]f(x) = x^3[/latex]
- Domain: All real numbers
- Range: All real numbers
- Reciprocal Function: [latex]f(x) = \frac{1}{x}[/latex]
- Domain: All real numbers except 0
- Range: All real numbers except 0
- Reciprocal Squared Function: [latex]f(x) = \frac{1}{x^2}[/latex]
- Domain: All real numbers except 0
- Range: [latex](0, \infty)[/latex]
- Square Root Function: [latex]f(x) = \sqrt{x}[/latex]
- Domain: [latex][0, \infty)[/latex]
- Range: [latex][0, \infty)[/latex]
- Cube Root Function: [latex]f(x) = \sqrt[3]{x}[/latex]
- Domain: All real numbers
- Range: All real numbers
Key Techniques
- Analyzing Domain:
- Consider restrictions (e.g., division by zero, even roots of negative numbers)
- Identify any x-values that produce undefined results
- Analyzing Range:
- Consider the function’s behavior for all valid inputs
- Identify any y-values that cannot be achieved
- Using Interval Notation:
- Use square brackets [ ] for inclusive endpoints
- Use parentheses ( ) for exclusive endpoints
- Use infinity symbols when there’s no upper or lower bound
- Graphical Analysis:
- Visualize the function to confirm domain and range
- Pay attention to asymptotes and end behavior
You can view the transcript for “1.2.h Domain and Range of Toolkit Functions” here (opens in new window).
Piecewise-Defined Functions
The Main Idea
- Definition:
- Functions defined by different formulas over different parts of the domain
- Notation uses curly braces and if-statements
- Absolute Value Function:
- Classic example of a piecewise function
- [latex]|x| = \begin{cases} x & \text{if } x \geq 0 \ -x & \text{if } x < 0 \end{cases}[/latex]
- Domain and Range:
- Domain is the union of all piece domains
- Range depends on the specific functions used
- Graphing:
- Combine graphs of individual pieces
- Pay attention to endpoints and continuity
Key Techniques
- Writing Piecewise Functions:
- Identify intervals for different rules
- Determine formulas for each interval
- Use proper notation with curly braces
- Evaluating Piecewise Functions:
- Determine which piece applies to the input
- Use the corresponding formula
- Graphing Piecewise Functions:
- Graph each piece on its interval
- Use open/closed circles for endpoints
- Ensure the function passes the vertical line test
You can view the transcript for “Ex 2: Graph a Piecewise Defined Function” here (opens in new window).