Functions
Explore domain and range, function transformations, composite functions, and inverse functions.
Domain & Range
A function is a rule that assigns exactly one output to each input. The set of all valid inputs is the domain; the set of all outputs is the range.
Domain
All possible x-values the function can accept. Check for: division by zero, square roots of negatives, and log of non-positive values.
Range
All possible y-values the function can produce. Determined by the behaviour of the function across its domain.
Example: f(x) = √(x − 3)
Domain: x − 3 ≥ 0 ⇒ x ≥ 3. Domain = [3, ∞)
Range: √(x − 3) ≥ 0. Range = [0, ∞)
Transformations of Functions
We can transform a function y = f(x) by translating, reflecting, or dilating its graph.
| Transformation | Equation | Effect |
|---|---|---|
| Translate up/down | y = f(x) + k | Shifts graph up (k>0) or down (k<0) |
| Translate left/right | y = f(x − h) | Shifts graph right (h>0) or left (h<0) |
| Reflect in x-axis | y = −f(x) | Flips graph vertically |
| Reflect in y-axis | y = f(−x) | Flips graph horizontally |
| Vertical dilation | y = a × f(x) | Stretches (a>1) or compresses (0<a<1) vertically |
| Horizontal dilation | y = f(bx) | Compresses (b>1) or stretches (0<b<1) horizontally |
Composite & Inverse Functions
Composite Functions
A composite function applies one function then another: (f ∘ g)(x) = f(g(x)).
Example: f(x) = 2x + 1, g(x) = x2
f(g(x)) = f(x2) = 2x2 + 1
g(f(x)) = g(2x+1) = (2x+1)2
Inverse Functions
The inverse f−1(x) “undoes” f. Its graph is the reflection of f in the line y = x.
To find f−1:
1. Write y = f(x)
2. Swap x and y
3. Solve for y
Example: Find the inverse of f(x) = 3x − 5
Step 1: y = 3x − 5
Step 2: Swap: x = 3y − 5
Step 3: Solve: x + 5 = 3y ⇒ y = (x + 5)/3
So f−1(x) = (x + 5)/3
Knowledge Check
Test your understanding of functions. Questions progress from easy to hard.
Question 1
What is the domain of f(x) = 1/(x − 2)?
Question 2
If f(x) = x2 + 3, what is the range of f?
Question 3
The graph of y = f(x) is shifted 4 units to the right. What is the new equation?
Question 4
If f(x) = 2x + 1 and g(x) = x2, what is f(g(3))?
Question 5
Which transformation describes y = −f(x)?
Question 6
Find the inverse of f(x) = 4x + 8.
Question 7
If f(x) = x2 and g(x) = x + 3, what is g(f(x))?
Question 8
The graph of y = f(x) is vertically stretched by a factor of 3 and then shifted up by 2. What is the new equation?
Question 9
If f(f−1(x)) = x for all x in the domain, what does this tell us?
Question 10
Given f(x) = √(2x + 6), find the domain and determine f−1(x).
Key Concepts Summary
- ●The domain is the set of valid inputs; the range is the set of outputs.
- ●Transformations: translations (h, k), reflections (−f(x), f(−x)), and dilations (af(x), f(bx)).
- ●Composite functions: f(g(x)) means apply g first, then f.
- ●Inverse functions undo each other: f(f−1(x)) = x. Find by swapping x and y.
- ●A function has an inverse only if it is one-to-one (passes the horizontal line test).