Functions

Summary and Exam Tips for Functions

Functions is a subtopic of Functions, which falls under the subject Mathematics in the IB DP curriculum. A function is a special type of relation where each element in set $A$ is associated with exactly one element in set $B$. This is denoted as $F: A \to B$. If an element in $A$ maps to multiple elements in $B$, it is not a function. The domain of a function is the set of all possible inputs, while the range is the set of all possible outputs. For example, the domain of $f(x) = \sin x$ is all real numbers, and its range is $[-1, 1]$.

Composite functions involve combining two functions, $f$ and $g$, to form a new function $h(x) = g(f(x))$. One-to-one (injective) functions map each element of $A$ to a unique element of $B$, while onto (surjective) functions cover every element in $B$. An inverse function reverses the mapping of a function, denoted as $f^{-1}$, and exists only for one-to-one functions. For instance, if $f(x) = 4x - 8$, its inverse is $f^{-1}(x) = \frac{x + 8}{4}$.

Exam Tips

• Understand Definitions: Make sure you grasp the definitions of functions, domain, range, and types of functions (one-to-one, onto).
• Practice Problems: Work through examples of finding domains and ranges, especially for functions involving fractions and square roots.
• Composite Functions: Practice forming and breaking down composite functions to understand their components.
• Inverse Functions: Learn to find inverse functions and verify them using $(f \circ f^{-1})(x) = x$.
• Graphical Understanding: Use graphs to visualize functions, especially to determine if they are one-to-one or onto.