1.1 Functions
1. Definition and Notation
A relation connects elements from a domain (set $X$) to a codomain (set $Y$).
A function $f$ from set $X$ to set $Y$ is a special relation where each element in set $X$ is mapped to one and only one element in set $Y$.
Notation: $f : x \mapsto y$ or $f(x) = y$, where:
- $x$ is the object (input, independent variable).
- $y$ or $f(x)$ is the image (output, dependent variable).
2. Types of Relations
- One-to-one relation: Each object has only one image, and each image belongs to only one object. (Is a function)
- Many-to-one relation: Two or more objects map to the same image. (Is a function)
- One-to-many relation: One object maps to more than one image. (NOT a function)
- Many-to-many relation: Multiple objects map to multiple images. (NOT a function)
3. Vertical Line Test
To determine whether a graph represents a function, draw a vertical line anywhere across the graph:
- If the vertical line intersects the graph at only one point, the graph is a function.
- If it intersects at more than one point, it is NOT a function.
4. Domain, Codomain, Range, and Undefined Values
For a function mapping set $X = \{1, 2, 3\}$ to set $Y = \{2, 4, 6, 8\}$ where $f(x) = 2x$:
- Domain: Set of all objects = $\{1, 2, 3\}$
- Codomain: Set of all possible outputs = $\{2, 4, 6, 8\}$
- Range: Set of actual images = $\{2, 4, 6\}$
Undefined Conditions (Excluded Values in Domain):
- For rational functions like $f(x) = \frac{a}{gx + h}$, the function is undefined when the denominator is zero ($gx + h = 0 \implies x = -\frac{h}{g}$). Domain restriction: $x \neq -\frac{h}{g}$.
- For square root functions like $f(x) = \sqrt{gx + h}$, the expression under the root must be non-negative ($gx + h \ge 0$).
5. Absolute Value Functions
An absolute value function is written as $f(x) = |g(x)|$, defined as:
$$|g(x)| = \begin{cases} g(x) & \text{if } g(x) \ge 0 \\ -g(x) & \text{if } g(x) < 0 \end{cases}$$
The absolute value ensures that output values $f(x)$ are always non-negative ($f(x) \ge 0$). Graphically, any part of the curve below the $x$-axis is reflected upwards across the $x$-axis (V-shaped graph at vertex points).
1.2 Composite Functions
1. Definition and Notation
When two functions $f$ and $g$ are combined such that $g$ operates first on $x$ and $f$ operates on the result $g(x)$, the combined operation is a composite function, written as $fg(x)$ or $(f \circ g)(x)$.
- $fg(x) = f(g(x))$: Process $g(x)$ first, then substitute into $f$.
- $gf(x) = g(f(x))$: Process $f(x)$ first, then substitute into $g$.
- $f^2(x) = ff(x) = f(f(x))$: Apply $f$ twice.
Note: In general, $fg(x) \neq gf(x)$ (composite operations are non-commutative).
2. Determining an Individual Function from a Composite Function
- Case A: Given $f(x)$ and $fg(x)$, find $g(x)$ (Finding inner function):
Substitute $g(x)$ directly into $f(x)$ expression and solve for $g(x)$.
- Case B: Given $g(x)$ and $fg(x)$, find $f(x)$ (Finding outer function):
Let $u = g(x)$, express $x$ in terms of $u$, and substitute into $fg(x)$ to find $f(u)$, then replace $u$ with $x$.
1.3 Inverse Functions
1. Definition and Properties
If a function $f$ maps $x$ to $y$ ($f(x) = y$), then its inverse function $f^{-1}$ maps $y$ back to $x$ ($f^{-1}(y) = x$).
- Existence Condition: A function $f$ has an inverse $f^{-1}$ if and only if $f$ is a one-to-one function.
- Horizontal Line Test: If any horizontal line intersects the graph of $f(x)$ at only one point, $f$ is a one-to-one function and $f^{-1}$ exists.
- Cancellation Property: $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.
- Domain and Range Swap:
$$\text{Domain of } f^{-1} = \text{Range of } f$$
$$\text{Range of } f^{-1} = \text{Domain of } f$$
- Symmetry: The graph of $y = f^{-1}(x)$ is the reflection of $y = f(x)$ along the line $y = x$.
2. Method to Find the Inverse Function $f^{-1}(x)$
- Set $y = f(x)$.
- Rearrange the equation algebraically to express $x$ as a subject in terms of $y$ ($x = g(y)$).
- Replace $x$ with $f^{-1}(y)$.
- Substitute $y$ back with $x$ to get $f^{-1}(x)$.