Summary
In this topic, you will learn about different types of functions, their properties, and how to graph them. You will also explore transformations and how to solve equations involving modulus functions.
- Modulus Function — The modulus of a number is its absolute value, which is always non-negative. Example: |x| = x if x >= 0 and |x| = -x if x < 0.
- One-One Function — A function where each element of the domain maps to a unique element in the range. Example: f(x) = x + 1 is one-one because each input gives a different output.
- Many-One Function — A function where multiple elements of the domain map to the same element in the range. Example: f(x) = x^2 is many-one because both -1 and 1 map to 1.
- Composite Function — A function created by applying one function to the results of another. Example: If f(x) = x^2 and g(x) = x + 2, then f(g(x)) = (x + 2)^2.
- Inverse Function — A function that reverses the effect of the original function. Example: If f(x) = 2x, then f⁻¹(x) = x/2.
- Transformation — Changing the position or size of a graph. Example: Translating a graph 2 units to the right.
- y = |f(x)| and y = f(|x|) — Graph transformations involving absolute values. Example: Reflecting the part of the graph below the x-axis.
Exam Tips
Key Definitions to Remember
- Modulus Function
- One-One Function
- Composite Function
- Inverse Function
Common Confusions
- Mixing up one-one and many-one functions
- Incorrectly finding the inverse of a non-one-one function
Typical Exam Questions
- What is the range of the function f(x) = x^2 + 1? The range is y ≥ 1.
- How do you find the inverse of f(x) = 3x + 2? Solve y = 3x + 2 for x to get f⁻¹(x) = (x - 2)/3.
- What are the solutions to |x + 2| = 3? x = 1 or x = -5.
What Examiners Usually Test
- Understanding and applying the modulus function
- Finding and verifying inverse functions
- Solving equations involving composite functions
- Graph transformations and their effects