Input: The value that goes into a function (also called the independent variable or x-value)
Output: The value that comes out of a function (also called the dependent variable or y-value)
Note: In function notation f(x), x is the input and f(x) is the output.
- Identify the function rule and variable
- Determine which value is being substituted (input)
- Calculate the result after substitution (output)
- Label the input and output values appropriately
f(x) = 2x + 3
Input: x = 5
Output: f(5) = 2(5) + 3 = 10 + 3 = 13
Input: 5, Output: 13
• Input identification: The value substituted into the function
• Output calculation: The result after applying the function rule
• Function evaluation: Replace x with the input value and simplify
• Practice Tip: Input values are always what you put INTO the function
- In g(x) = x + 7 when x = 3: Input = 3, Output = 10
- In h(x) = 4x - 1 when x = 2: Input = 2, Output = 7
- In p(x) = -x + 5 when x = -2: Input = -2, Output = 7
- Input values are always the values substituted into the function
- Output values are always the results after applying the function rule
- Think of a function as a machine: inputs go in, outputs come out
Q: What's the difference between input and output?
A: Input is what you put INTO the function, output is what comes OUT of the function.
Q: Can I call input values x-values?
A: Yes, input values are often called x-values or independent variables.
Function table: A table showing input values and their corresponding output values
Note: Function tables help visualize the relationship between inputs and outputs of a function.
- 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
| Input (x) | Output f(x) = x² - 1 |
|---|---|
| -2 | 3 |
| -1 | 0 |
| 0 | -1 |
| 1 | 0 |
| 2 | 3 |
The function table is completed with 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: Work through each value systematically to avoid missing any
- 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.
Domain: The set of all possible input values (x-values) of a function
Range: The 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, 3), (2, 5), (3, 7), (4, 9), the inputs are: 1, 2, 3, 4
Domain = {1, 2, 3, 4}
From (1, 3), (2, 5), (3, 7), (4, 9), the outputs are: 3, 5, 7, 9
Range = {3, 5, 7, 9}
Domain = {1, 2, 3, 4}, Range = {3, 5, 7, 9}
• 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.
Ordered pair: A pair of numbers (x, y) where x is the input and y is the output
Note: For a relation to be a function, each input (x-value) must have exactly one output (y-value).
- 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
Set 1: Each input has exactly one output → It's a function
Set 2: Input 1 has two outputs → It's not a function
The set {(1, 2), (2, 3), (3, 4), (4, 5)} represents 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.
Input-output relationship: The connection between inputs and outputs in a function
Note: Sometimes we know the output and need to find the input, which requires solving an equation.
- Set the function equal to the known output value
- Solve the resulting equation for the input value
- Verify the solution by substituting back into the function
Since f(x) = 10, we have: 3x - 2 = 10
3x - 2 = 10
3x = 10 + 2
3x = 12
x = 4
Check: f(4) = 3(4) - 2 = 12 - 2 = 10 ✓
The input value is x = 4
• Equation setup: Set function equal to known output
• Solving: Use inverse operations to isolate the variable
• Verification: Substitute solution back to confirm
• Practice Tip: Always check your answer by substituting back into the original function
- If g(x) = 2x + 5 and g(x) = 11, then 2x + 5 = 11, so x = 3
- If h(x) = x - 7 and h(x) = 2, then x - 7 = 2, so x = 9
- If p(x) = -x + 4 and p(x) = 1, then -x + 4 = 1, so x = 3
- When finding input from output, set the function equal to the output and solve
- Use inverse operations to solve for the input variable
- Always verify your answer by substituting back into the function
Q: What if I have a quadratic function?
A: You'll get a quadratic equation to solve, which might have zero, one, or two solutions.
Q: Can there be more than one input for the same output?
A: Yes, in many functions multiple inputs can have the same output.