Function: A relation where each input (domain element) has exactly one output (range element)
Note: A function cannot have the same input paired with multiple outputs. This is known as the vertical line test for graphs.
- Examine the input values (first coordinates) in each ordered pair
- Check if any input value appears more than once
- If an input has multiple outputs, it's not a function
- If each input has exactly one output, it is a function
Inputs: {1, 2, 3, 4} - Each input appears only once
Outputs: {2, 3, 4, 5} - Each input has exactly one output
Inputs: {1, 1, 2, 3} - Input 1 appears twice
Outputs: {2, 3, 4, 5} - Input 1 maps to both 2 and 3
Relation 1: Each input has exactly one output → It's a function
Relation 2: Input 1 has two outputs → It's not a function
The relation {(1, 2), (2, 3), (3, 4), (4, 5)} is a function because each input has exactly one output.
• Function definition: Each input must have exactly one output
• No repeated inputs: Same input cannot map to different outputs
• Verification: Check each input value for uniqueness
• Practice Tip: Create a table with inputs and outputs to visualize the relationship
- {(a, 1), (b, 2), (c, 3)} - Function (different inputs)
- {(x, 5), (y, 5), (z, 5)} - Function (same output is OK)
- {(2, 4), (2, 6), (3, 7)} - Not a function (input 2 has two outputs)
- Focus on the inputs (domain) when determining if a relation is a function
- Multiple inputs can have the same output - that's still a function
- Only inputs matter - the same output for different inputs is allowed
Q: Can different inputs have the same output in a function?
A: Yes, absolutely. For example, f(x) = x² has f(2) = 4 and f(-2) = 4.
Q: What if I have an empty relation?
A: An empty relation is technically a function since there are no violations of the function rule.
Domain: Set of all possible input values (x-values) of a function
Range: Set of all possible output values (y-values) of a function
Note: For a finite set of ordered pairs, the domain is the set of all first coordinates, and the range is the set of all second coordinates.
- Identify all input values (first coordinates) in the ordered pairs
- List the unique input values to form the domain
- Identify all output values (second coordinates) in the ordered pairs
- List the unique output values to form the range
From (1, 5), (2, 7), (3, 9), (4, 11), the inputs are: 1, 2, 3, 4
Domain = {1, 2, 3, 4}
From (1, 5), (2, 7), (3, 9), (4, 11), the outputs are: 5, 7, 9, 11
Range = {5, 7, 9, 11}
Domain = {1, 2, 3, 4}, Range = {5, 7, 9, 11}
• Domain: Collect all first coordinates of ordered pairs
• Range: Collect all second coordinates of ordered pairs
• Uniqueness: Each value listed only once in sets
• Practice Tip: Organize data in a table to clearly separate inputs and outputs
- For {(a, x), (b, y), (c, z)}, Domain = {a, b, c}, Range = {x, y, z}
- For {(0, 1), (1, 1), (2, 1)}, Domain = {0, 1, 2}, Range = {1}
- For {(−1, 0), (0, 1), (1, 0)}, Domain = {−1, 0, 1}, Range = {0, 1}
- Domain values are always the inputs (x-values or first coordinates)
- Range values are always the outputs (y-values or second coordinates)
- Same output values don't need to be repeated in the range set
Q: What if the same output appears multiple times?
A: Only list it once in the range set. For example, if outputs are {3, 5, 3, 7}, the range is {3, 5, 7}.
Q: Can the domain and range be infinite sets?
A: Yes, for functions defined over all real numbers, the domain and/or range may be infinite.
Function notation: f(x) represents the output of function f when input is x
Note: f(x) is read as "f of x" and represents the y-value or output corresponding to input x.
- Identify the function rule (the expression after f(x) = )
- Replace every instance of x with the given input value
- Simplify the expression using order of operations
- Record the result as the function output
f(4) = 3(4) - 2 = 12 - 2 = 10
f(-1) = 3(-1) - 2 = -3 - 2 = -5
f(4) = 10 and f(-1) = -5
f(4) = 10 and f(-1) = -5
• Substitution: Replace x with the given input value
• Order of operations: Perform operations in correct sequence
• Simplification: Calculate the final result accurately
• Practice Tip: Always perform operations in the correct order: parentheses, exponents, multiplication/division, addition/subtraction
- If g(x) = x² + 1, then g(3) = 3² + 1 = 10
- If h(x) = 2x - 5, then h(0) = 2(0) - 5 = -5
- If p(x) = -x + 4, then p(-2) = -(-2) + 4 = 6
- Think of f(x) as a machine that takes input x and produces output f(x)
- Always substitute the input value in place of every occurrence of x
- Be careful with negative inputs and negative coefficients
Q: What does f(x) mean?
A: f(x) means "the value of function f when x is the input" - it's another way of writing y.
Q: Can I have multiple functions like f(x), g(x), h(x)?
A: Yes, different letters represent different functions: f(x), g(x), h(x), etc.
Input-output table: A table showing the relationship between inputs (x) and outputs (f(x)) of a function
Note: These tables help visualize how a function transforms inputs into outputs and identify patterns in the function behavior.
- Set up the table with input (x) and output (f(x)) columns
- For each input value, substitute into the function rule
- Calculate the corresponding output value
- Fill in the table with the ordered pairs (x, f(x))
f(-2) = (-2)² - 1 = 4 - 1 = 3
f(-1) = (-1)² - 1 = 1 - 1 = 0
f(0) = (0)² - 1 = 0 - 1 = -1
f(1) = (1)² - 1 = 1 - 1 = 0
f(2) = (2)² - 1 = 4 - 1 = 3
| x (input) | f(x) = x² - 1 (output) |
|---|---|
| -2 | 3 |
| -1 | 0 |
| 0 | -1 |
| 1 | 0 |
| 2 | 3 |
The completed table shows the ordered pairs: (-2, 3), (-1, 0), (0, -1), (1, 0), (2, 3)
• Systematic substitution: Substitute each input value into the function rule
• Order of operations: Apply operations in correct sequence
• Organization: Keep inputs and outputs aligned in table format
• Practice Tip: Verify your calculations by checking a few values
- For g(x) = 2x + 1 and x ∈ {0, 1, 2}: (0, 1), (1, 3), (2, 5)
- For h(x) = -x² and x ∈ {-1, 0, 1}: (-1, -1), (0, 0), (1, -1)
- For p(x) = x³ and x ∈ {-1, 0, 1}: (-1, -1), (0, 0), (1, 1)
- Organize your work in a table to avoid missing any input values
- Be careful with negative numbers raised to even and odd powers
- Look for patterns in the output values to verify your calculations
Q: What if I have many input values?
A: Work systematically through each value, and consider using a calculator for complex functions.
Q: Can I use the table to graph the function?
A: Yes, the ordered pairs (x, f(x)) are the points you plot on the coordinate plane.
Function evaluation: Finding the output value of a function for a specific input value
Note: This involves substituting the input value for x in the function rule and simplifying using order of operations.
- Identify the function rule: f(x) = 2x² - 3x + 1
- For each input value, substitute into the function rule
- Follow order of operations: exponents, multiplication, addition/subtraction
- Calculate and record each output value
f(0) = 2(0)² - 3(0) + 1 = 2(0) - 0 + 1 = 0 - 0 + 1 = 1
f(1) = 2(1)² - 3(1) + 1 = 2(1) - 3 + 1 = 2 - 3 + 1 = 0
f(-2) = 2(-2)² - 3(-2) + 1 = 2(4) + 6 + 1 = 8 + 6 + 1 = 15
f(0) = 1, f(1) = 0, f(-2) = 15
• Substitution: Replace every x with the given input value
• Order of operations: Exponents first, then multiplication, then addition/subtraction
• Sign handling: Be careful with negative inputs, especially with exponents
• Practice Tip: Work slowly and carefully, especially with multiple operations
- If g(x) = x² + 2x, then g(3) = 9 + 6 = 15
- If h(x) = -x² + 4, then h(-1) = -1 + 4 = 3
- If p(x) = 3x² - x + 2, then p(0) = 0 - 0 + 2 = 2
- Always substitute the input value in place of every occurrence of x
- Pay special attention to the sign when raising negative numbers to powers
- Work step by step to avoid calculation errors
Q: What if the function has multiple terms with x?
A: Substitute the input value for every occurrence of x, then calculate each term separately.
Q: How do I handle negative inputs with exponents?
A: (-a)² = a² (positive), but (-a)³ = -a³ (negative). The result depends on whether the exponent is even or odd.