What is a Function?
A function is a special relationship between two quantities where each input (x-value) produces exactly one output (y-value). This is called the vertical line test: if you can draw a vertical line anywhere on the graph and it only touches the graph once, then the relation is a function.
Think of a function as a machine: you put an input in one end, and the machine processes it according to a rule and produces exactly one output.
Function Notation
We write f(x) to denote "f of x," meaning the function f evaluated at x:
This means "the function f takes x, doubles it, and adds 3."
Example 1: Evaluating Functions
If f(x) = 2x + 3, find:
f(2) = 2(2) + 3 = 4 + 3 = 7
f(-1) = 2(-1) + 3 = -2 + 3 = 1
f(0) = 2(0) + 3 = 3
Domain and Range
Domain
The domain is the set of all possible input values (x-values) for which the function is defined.
Common Domain Restrictions
- Denominators: Cannot be zero → exclude values that make denominator = 0
- Square roots (even roots): radicand must be ≥ 0
- Logarithms: argument must be > 0
Example 2: Finding Domain
Find the domain of f(x) = 1/(x - 2)
Denominator cannot be zero: x - 2 ≠ 0
x ≠ 2
Domain = all real numbers except 2, or (-∞, 2) ∪ (2, ∞)
Example 3: Domain with Square Root
Find the domain of f(x) = √(x - 4)
Square root requires: x - 4 ≥ 0
x ≥ 4
Domain = [4, ∞)
Range
The range is the set of all possible output values (y-values) that the function can produce.
Example 4: Finding Range
Find the range of f(x) = x² with domain all real numbers
x² is always ≥ 0
Range = [0, ∞)
Types of Functions
Linear Functions
A linear function has the form f(x) = mx + b, where m is the slope and b is the y-intercept. Its graph is a straight line.
Example 5: Linear Function
f(x) = 3x - 2
Slope = 3, Y-intercept = -2
This line rises 3 units for every 1 unit it runs right, and crosses the y-axis at -2.
Quadratic Functions
A quadratic function has the form f(x) = ax² + bx + c, where a ≠ 0. Its graph is a parabola.
Vertex Form
where (h, k) is the vertex of the parabola.
If a > 0, the parabola opens upward (minimum at vertex)
If a < 0, the parabola opens downward (maximum at vertex)
Example 6: Converting to Vertex Form
Convert f(x) = x² - 6x + 8 to vertex form
Complete the square:
f(x) = (x² - 6x + 9) + 8 - 9
f(x) = (x - 3)² - 1
Vertex = (3, -1)
Polynomial Functions
Polynomial functions are sums of terms of the form axⁿ, where n is a non-negative integer.
- Degree 0: Constant (f(x) = c)
- Degree 1: Linear
- Degree 2: Quadratic
- Degree 3: Cubic
Absolute Value Function
f(x) = |x| creates a V-shaped graph
Rational Functions
Rational functions are ratios of polynomials: f(x) = P(x)/Q(x)
They have vertical asymptotes where the denominator equals zero.
Function Operations
Arithmetic Operations
Given f(x) and g(x), we can create new functions:
Example 7: Function Addition
If f(x) = x² and g(x) = 2x + 1, find (f + g)(3)
(f + g)(3) = f(3) + g(3) = 9 + 7 = 16
Or: (f + g)(x) = x² + 2x + 1 = (x + 1)²
(f + g)(3) = (3 + 1)² = 16 ✓
Composition of Functions
Function composition applies one function to the result of another:
Read as "f composed with g" or "f of g of x"
Example 8: Function Composition
If f(x) = 2x + 3 and g(x) = x², find f(g(2))
Step 1: g(2) = 2² = 4
Step 2: f(4) = 2(4) + 3 = 11
f(g(2)) = 11
Example 9: Finding Composite Functions
If f(x) = 2x + 3 and g(x) = x², find f(g(x))
f(g(x)) = f(x²) = 2(x²) + 3 = 2x² + 3
g(f(x)) = g(2x + 3) = (2x + 3)² = 4x² + 12x + 9
Note: f(g(x)) ≠ g(f(x)) in general!
Inverse Functions
An inverse function f⁻¹(x) "undoes" what f(x) does. If f maps x to y, then f⁻¹ maps y back to x.
The domain and range of f become the range and domain of f⁻¹, respectively.
Example 10: Finding an Inverse Function
Find the inverse of f(x) = (x - 2)/3
Step 1: Write y = (x - 2)/3
Step 2: Swap x and y: x = (y - 2)/3
Step 3: Solve for y:
3x = y - 2
y = 3x + 2
f⁻¹(x) = 3x + 2
Horizontal Line Test
A function has an inverse if and only if it passes the horizontal line test—no horizontal line intersects the graph more than once. This means the function must be one-to-one.
Function Transformations
Transformations shift, stretch, or reflect the graph of a function.
Vertical Transformations
- y = f(x) + k: Shift up by k (down if k < 0)
- y = k·f(x): Vertical stretch by factor k (reflect if k < 0)
Horizontal Transformations
- y = f(x - h): Shift right by h (left if h < 0)
- y = f(kx): Horizontal compression by factor k (stretch if 0 < k < 1)
Example 11: Transformations
Starting with f(x) = x², describe the transformations for g(x) = 3(x - 2)² + 1
- (x - 2): Shift right 2 units
- ³ outside: Vertical stretch by factor 3
- + 1: Shift up 1 unit
The vertex moves from (0, 0) to (2, 1).
Even and Odd Functions
Even function: f(-x) = f(x) — symmetric about the y-axis
Odd function: f(-x) = -f(x) — symmetric about the origin
Example 12: Even vs Odd
f(x) = x² is even: f(-2) = 4 = f(2)
f(x) = x³ is odd: f(-2) = -8 = -f(2)
f(x) = x² + x is neither: f(-2) = 2 + (-2) = 0, but f(2) = 6