Function Table: A table that lists input values (domain) and their corresponding output values (range) for a function
- Identify the function rule
- List the input values in the first column
- Substitute each input into the function rule
- Calculate the corresponding output for each input
- Record the ordered pairs (input, output)
- Verify calculations for accuracy
Given: f(x) = 2x + 3
This means "multiply input by 2, then add 3"
Input values: -2, -1, 0, 1, 2
f(-2) = 2(-2) + 3 = -4 + 3 = -1
f(-1) = 2(-1) + 3 = -2 + 3 = 1
f(0) = 2(0) + 3 = 0 + 3 = 3
f(1) = 2(1) + 3 = 2 + 3 = 5
f(2) = 2(2) + 3 = 4 + 3 = 7
| Input (x) | Output f(x) |
|---|---|
| -2 | -1 |
| -1 | 1 |
| 0 | 3 |
| 1 | 5 |
| 2 | 7 |
The function table for f(x) = 2x + 3 with inputs {-2, -1, 0, 1, 2} shows the corresponding outputs {-1, 1, 3, 5, 7}.
• Function Evaluation: Substitute input values into function rule
• Order of Operations: Follow PEMDAS when calculating
• Ordered Pairs: Represent as (input, output)
Function Property: Each input value (x) must correspond to exactly one output value (y)
Input values: -1, 0, 1, 2, 3
Each input appears exactly once in the table
-1 appears once, 0 appears once, 1 appears once, 2 appears once, 3 appears once
Input -1 → Output 2 (only one output)
Input 0 → Output 4 (only one output)
Input 1 → Output 6 (only one output)
Input 2 → Output 4 (only one output)
Input 3 → Output 2 (only one output)
Since each input corresponds to exactly one output, this table represents a function
It's perfectly fine for different inputs to have the same output (0 and 2 both map to 4)
The restriction is only on inputs having multiple outputs
Domain: {-1, 0, 1, 2, 3}
Range: {2, 4, 6}
Yes, this table represents a function because each input value has exactly one corresponding output value.
• Function Property: Each input maps to exactly one output
• Input Uniqueness: Check that no input appears more than once
• Output Variation: Different inputs can have same output
Function Rule: The mathematical expression that describes how to transform input values into output values
(0, 5), (1, 8), (2, 11), (3, 14), (4, 17)
5 → 8 → 11 → 14 → 17
Each output increases by 3 from the previous output
When x increases by 1, y increases by 3
Rate of change = Δy/Δx = 3/1 = 3
When x = 0, y = 5
This is the y-intercept (b in y = mx + b)
Since rate of change = 3 and y-intercept = 5
f(x) = 3x + 5
Check with point (1, 8): f(1) = 3(1) + 5 = 8 ✓
Check with point (2, 11): f(2) = 3(2) + 5 = 11 ✓
Check with point (3, 14): f(3) = 3(3) + 5 = 14 ✓
Check with point (4, 17): f(4) = 3(4) + 5 = 17 ✓
The function rule for the table is f(x) = 3x + 5.
• Pattern Recognition: Look for consistent changes in outputs
• Rate of Change: Calculate slope from consecutive points
• Y-Intercept: Identify output when input is zero
Function Table: A table organizing input-output pairs for a function
Domain: The set of all possible input values for a function
Range: The set of all possible output values for a function
Input: The independent variable (x) that goes into a function
Output: The dependent variable (y or f(x)) that comes out of a function
Ordered Pair: A pair (x, y) representing an input-output relationship
Function Rule: The mathematical expression that defines the relationship between inputs and outputs
Linear Function: A function with a constant rate of change, graphing as a straight line
Rate of Change: The change in output divided by the change in input
- Examine the table: Look at the structure and organization
- Identify the relationship: Determine if it's a function
- Find patterns: Look for consistent changes in outputs
- Calculate rate of change: Determine if it's constant
- Find the rule: Derive the function rule
- Verify: Check that the rule fits all given points
• Function Property: Each input maps to exactly one output
• Domain: Set of all input values
• Range: Set of all output values
• Linear Pattern: Constant rate of change indicates linear function
• Verification: Always check rule against given points
Real-World Function: A function that models a practical scenario with meaningful input and output values
Let d = number of days rented
Fixed cost per day: $25
Mileage cost per day: $0.15 × 100 miles = $15 per day
Total cost = (Daily rate × Days) + (Mileage rate × Miles per day × Days)
C(d) = 25d + 0.15(100d) = 25d + 15d = 40d
C(1) = 40(1) = $40
C(2) = 40(2) = $80
C(3) = 40(3) = $120
| Days (d) | Cost C(d) |
|---|---|
| 1 | $40 |
| 2 | $80 |
| 3 | $120 |
Each input (days) corresponds to exactly one output (cost)
This confirms the relationship is a function
The slope is 40, meaning cost increases by $40 per day
The function rule is C(d) = 40d. The cost for 1 day is $40, for 2 days is $80, and for 3 days is $120.
• Real-World Modeling: Translate scenario into mathematical function
• Cost Analysis: Separate fixed and variable components
• Linear Function: Constant rate of change
Quadratic Function: A function of degree 2 with the general form f(x) = ax² + bx + c
Given: f(x) = x² - 2x + 1
This is a quadratic function (highest power of x is 2)
f(-1) = (-1)² - 2(-1) + 1 = 1 + 2 + 1 = 4
f(0) = (0)² - 2(0) + 1 = 0 - 0 + 1 = 1
f(1) = (1)² - 2(1) + 1 = 1 - 2 + 1 = 0
f(2) = (2)² - 2(2) + 1 = 4 - 4 + 1 = 1
f(3) = (3)² - 2(3) + 1 = 9 - 6 + 1 = 4
| Input (x) | Output f(x) |
|---|---|
| -1 | 4 |
| 0 | 1 |
| 1 | 0 |
| 2 | 1 |
| 3 | 4 |
Notice: Differences between outputs are not constant (3, -1, 1, 3), confirming this is not linear
This is a quadratic function. The completed table shows outputs {4, 1, 0, 1, 4} for inputs {-1, 0, 1, 2, 3}.
• Quadratic Evaluation: Substitute values and follow order of operations
• Non-Linear Pattern: Variable differences indicate non-linear function
• Function Verification: Each input maps to exactly one output
Function Table: A tabular representation of a function showing input values and their corresponding output values
Domain: The set of all possible input values for which the function is defined
Range: The set of all possible output values that the function can produce
Function Rule: The mathematical expression that defines how to transform inputs into outputs
Linear Function: A function with a constant rate of change, represented by f(x) = mx + b
Quadratic Function: A function of degree 2, represented by f(x) = ax² + bx + c
Rate of Change: The ratio of the change in output to the change in input
Ordered Pair: A pair (x, y) representing an input-output relationship
- Identify the function rule: Recognize the formula that defines the function
- Substitute input values: Replace x with each given input
- Calculate outputs: Apply the function rule to find corresponding outputs
- Organize systematically: Place inputs and outputs in table format
- Identify patterns: Look for consistent changes in outputs
- Verify function property: Ensure each input maps to exactly one output
• Function Property: Each input maps to exactly one output
• Domain: Set of all possible input values
• Range: Set of all possible output values
• Linear Detection: Constant first differences indicate linear function
• Verification: Check function rule against all given points
Linear: f(x) = 2x + 1 (constant rate of change)
Quadratic: g(x) = x² (changing rate of change)
Cubic: h(x) = x³ (accelerating rate of change)
Tables show how different functions produce different output patterns
Analysis: The chart demonstrates how different function types create distinct patterns in their input-output relationships.
- Linear: Constant rate of change (arithmetic sequence in outputs)
- Quadratic: Changing rate of change (outputs change at increasing/decreasing rate)
- Cubic: Accelerating rate of change (rapidly changing outputs)
- Pattern recognition helps identify function type