Study Notes
Functions are mathematical expressions that relate an input to an output, often using the notation f(x). They are not multiplications of f and x.
- Function Notation — a way to write functions, typically as f(x), which represents the y-value in a function. Example: f(x) = 3x - 5 means for x = 4, f(4) = 7.
- Composite Functions — created when one function is substituted into another, written as f(g(x)). Example: If f(x) = 3x^2 + 2x + 1 and g(x) = 4x - 5, then f(g(x)) is calculated by substituting g(x) into f(x).
- Reciprocal Functions — functions of the form 1/f(x), which have vertical and horizontal asymptotes. Example: For f(x) = 1/(x+3), the domain excludes x = -3, and the range excludes y = 0.
Exam Tips
Key Definitions to Remember
- Function Notation: f(x) represents the output of a function for input x.
- Composite Function: f(g(x)) is the result of substituting g(x) into f(x).
- Reciprocal Function: 1/f(x) has vertical and horizontal asymptotes.
Common Confusions
- Misinterpreting f(x) as multiplication of f and x.
- Forgetting to use parentheses when evaluating functions.
Typical Exam Questions
- What is f(4) for f(x) = 3x - 5? f(4) = 7
- How do you find (f∘g)(x) for f(x) = x^2 + 6 and g(x) = 2x - 1? Substitute g(x) into f(x) to get 4x^2 - 4x + 7
- What is the domain of y = 1/(x+3)? All real numbers except x = -3
What Examiners Usually Test
- Understanding and using function notation correctly
- Ability to evaluate functions at given points
- Calculating composite functions and understanding their notation
- Identifying domains and ranges of reciprocal functions