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Function notation, domain, range, and evaluation
Learn step-by-step with practice exercises built right in.
A function is a special relationship between inputs and outputs where each input has exactly ONE output.
Think of a function as a machine:
Real-life examples:
Instead of writing y = 2x + 3, we can write:
f(x) = 2x + 3
Read as: "f of x equals 2x plus 3"
Parts:
If f(x) = 3x + 2, find f(5)
Step 1: Understand what f(5) means: It means substitute x = 5 into the function
Step 2: Replace x with 5 in the function: f(5) = 3(5) + 2
Step 3: Calculate: f(5) = 15 + 2 f(5) = 17
Answer: f(5) = 17
If , find
Avoid these 3 frequent errors
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Solve .
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Other function names:
Evaluating a function means finding the output for a specific input.
Example 1: If f(x) = 3x + 5, find f(2)
Substitute x = 2: f(2) = 3(2) + 5 f(2) = 6 + 5 f(2) = 11
This means when x = 2, the output is 11.
Example 2: If g(x) = x² - 4, find g(-3)
g(-3) = (-3)² - 4 g(-3) = 9 - 4 g(-3) = 5
Example 3: If h(x) = 2x² + x - 1, find h(0)
h(0) = 2(0)² + 0 - 1 h(0) = -1
Example 4: If f(x) = x² + 3x, find f(a)
f(a) = a² + 3a
We just replace x with a!
Example 1: If f(x) = x² + 1, find f(x + 2)
Substitute (x + 2) for every x: f(x + 2) = (x + 2)² + 1 f(x + 2) = x² + 4x + 4 + 1 f(x + 2) = x² + 4x + 5
Example 2: If g(x) = 2x - 3, find g(3a)
g(3a) = 2(3a) - 3 g(3a) = 6a - 3
Example 3: If h(x) = x², find h(x + h)
h(x + h) = (x + h)² h(x + h) = x² + 2xh + h²
The Vertical Line Test: A graph represents a function if NO vertical line crosses it more than once.
Example: Functions
Example: Not Functions
These fail because one x-value gives multiple y-values.
A table represents a function if each input appears only ONCE.
Function: ✓
| x | y |
|---|---|
| 1 | 3 |
| 2 | 5 |
| 3 | 7 |
Each x has exactly one y.
Not a function: ✗
| x | y |
|---|---|
| 1 | 3 |
| 1 | 5 |
| 2 | 7 |
x = 1 has two different outputs!
A set of ordered pairs is a function if no two pairs have the same first element with different second elements.
Function: {(1, 2), (2, 4), (3, 6)} ✓
Not a function: {(1, 2), (1, 3), (2, 5)} ✗ Because (1, 2) and (1, 3) have same input, different outputs.
A mapping diagram shows inputs pointing to outputs.
Function: Each input has exactly ONE arrow going out.
Not a function: If any input has MULTIPLE arrows going to different outputs.
Independent variable (input):
Dependent variable (output):
Example: In f(x) = 2x + 3
Linear Function: f(x) = mx + b Graph is a straight line Example: f(x) = 2x + 3
Quadratic Function: f(x) = ax² + bx + c Graph is a parabola Example: f(x) = x² - 4
Absolute Value Function: f(x) = |x| Graph is V-shaped Example: f(x) = |x - 2|
Constant Function: f(x) = c (always same output) Graph is horizontal line Example: f(x) = 5
Identity Function: f(x) = x (output equals input) Graph is line through origin with slope 1
Sometimes we know f(x) and need to find x.
Example 1: If f(x) = 3x - 2 and f(x) = 7, find x
Set up equation: 3x - 2 = 7 3x = 9 x = 3
So f(3) = 7
Example 2: If g(x) = x² and g(x) = 16, find x
x² = 16 x = ±4
Both x = 4 and x = -4 give output 16.
Example 3: If h(x) = 2x + 1 and h(x) = -3, find x
2x + 1 = -3 2x = -4 x = -2
Domain: The set of all possible INPUT values (x-values)
Range: The set of all possible OUTPUT values (y-values or f(x)-values)
Example 1: f(x) = x + 5 Domain: all real numbers Range: all real numbers
Example 2: g(x) = x² Domain: all real numbers Range: y ≥ 0 (squares are never negative)
Example 3: h(x) = 1/x Domain: all real numbers except 0 (can't divide by 0) Range: all real numbers except 0
Example 1: Temperature Conversion C(F) = (5/9)(F - 32)
Input: Fahrenheit temperature Output: Celsius temperature
C(32) = (5/9)(32 - 32) = 0°C C(212) = (5/9)(212 - 32) = 100°C
Example 2: Cost Function C(n) = 50 + 10n
Input: number of items (n) Output: total cost in dollars
C(5) = 50 + 10(5) = 100.
Example 3: Distance d(t) = 60t
Input: time in hours Output: distance in miles (at 60 mph)
d(3) = 60(3) = 180 miles After 3 hours, you've traveled 180 miles.
Composition means putting one function inside another.
Notation: (f ∘ g)(x) = f(g(x))
Read as: "f of g of x"
Example: If f(x) = x + 1 and g(x) = 2x
f(g(x)) = f(2x) = 2x + 1
First apply g, then apply f to the result.
Functions are everywhere in mathematics and real life:
Understanding functions is key to advanced mathematics!
Confusing f(x) with f · x f(x) is NOT f times x, it's function notation
Adding instead of substituting f(2) means replace x with 2, not add 2
Not using parentheses If f(x) = x², then f(2x) = (2x)² = 4x², not 2x²
Thinking every relation is a function Remember: each input needs exactly ONE output
Mixing up domain and range Domain = inputs (x), Range = outputs (y)
Method 1: Solve for y If you can solve for y and get only one y for each x, it's a function.
Example: x + y = 5 Solve: y = 5 - x ✓ Function (one y for each x)
Example: x² + y² = 25 Solve: y = ±√(25 - x²) ✗ Not a function (two y-values)
Method 2: Graph and use vertical line test
Method 3: Make a table and check for repeated inputs
Why use f(x) instead of y?
Names multiple functions clearly f(x), g(x), h(x) vs. y₁, y₂, y₃
Shows input explicitly f(3) clearly means input is 3
Easier composition f(g(x)) is clearer than nested y's
Better for applications C(n), d(t), P(x) tell you what the variables mean
When evaluating f(a):
When finding x given f(x):
Function Definition: Each input has exactly ONE output
Notation: f(x) = expression
Evaluation: f(a) means substitute a for x
Vertical Line Test: Function if no vertical line crosses graph twice
Domain: All possible inputs
Range: All possible outputs
Level 1: Simple evaluation
Level 2: Quadratic functions
Level 3: Expressions as inputs
Level 4: Working backwards
Level 5: Applications
To find , substitute into the function:
Answer:
Is this relation a function? {(1, 3), (2, 5), (3, 7), (4, 5)}
Step 1: Recall the definition of a function: A function is a relation where each input (x-value) has exactly ONE output (y-value).
Step 2: Check each x-value: x = 1 → y = 3 x = 2 → y = 5 x = 3 → y = 7 x = 4 → y = 5
Step 3: Determine if any x-value repeats: All x-values (1, 2, 3, 4) are different. Each x-value maps to only one y-value.
Note: It's okay for different x-values to have the same y-value (both 2 and 4 give 5).
Answer: Yes, this is a function
Given , find
Substitute into the function:
If g(x) = x² - 4x + 1, find g(-2)
Step 1: Substitute x = -2 into the function: g(-2) = (-2)² - 4(-2) + 1
Step 2: Calculate the squared term: (-2)² = 4
Step 3: Calculate the linear term: -4(-2) = 8
Step 4: Add all terms: g(-2) = 4 + 8 + 1 g(-2) = 13
Answer: g(-2) = 13
Is this relation a function? {(2, 3), (4, 5), (2, 7), (6, 9)}
Step 1: Check each x-value in the ordered pairs: (2, 3) → x = 2 maps to y = 3 (4, 5) → x = 4 maps to y = 5 (2, 7) → x = 2 maps to y = 7 ← Problem! (6, 9) → x = 6 maps to y = 9
Step 2: Identify the issue: The x-value 2 appears twice:
Step 3: Apply the definition: A function requires each input to have exactly ONE output. Since x = 2 has TWO different outputs (3 and 7), this violates the definition of a function.
Answer: No, this is NOT a function
Find the domain of
The domain is all real numbers except where the denominator equals zero.
Set the denominator equal to zero:
We cannot divide by zero, so must be excluded.
If h(x) = 2x - 5, find the value of x when h(x) = 11
Step 1: Set up the equation: We want h(x) = 11, so: 2x - 5 = 11
Step 2: Solve for x (add 5 to both sides): 2x = 11 + 5 2x = 16
Step 3: Divide both sides by 2: x = 8
Step 4: Check by finding h(8): h(8) = 2(8) - 5 = 16 - 5 = 11 ✓
This means when we input x = 8, the output is 11.
Answer: x = 8
Answer:
Answer: Domain: all real numbers except
In interval notation: