Understanding Functions: Domain, Range, and Types

Functions

• A function $f$ is a rule that assigns each element $x$ in a set $D$ to exactly one element $f(x)$ in a set $E$.
• The domain of the function is the set $D$.
• The range of $f$ is the set of all possible values of $f(x)$ as $x$ varies throughout the domain.
• The independent variable is $x$.
• The dependent variable is $y$.

Representing Functions

Functions can be represented in four ways:

• Verbally (by description in words)
• Visually (by a graph)
• Numerically (by a table of values)
• Algebraically (by an explicit formula)

Vertical Line Test

• A curve in the $xy$-plane is the graph of a function of $x$ if and only if no vertical line intersects the curve more than once.

Piecewise Defined Functions

• Functions can be defined differently on different intervals of their domain
• Example: $$ f(x) = \begin{cases} x^2 + 2, & \text{if } x < 0 \ x - 3, & \text{if } x \geq 0 \end{cases} $$

Symmetry of Functions

• For even functions, $f(-x) = f(x)$, indicating symmetry about the y-axis.
• For odd functions, $f(-x) = -f(x)$, indicating symmetry about the origin.

Types of Functions

Common Function types include:

• Linear Functions: $f(x) = mx + b$
• Power Functions: $f(x) = x^n$
• Polynomials: $f(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0$
• Rational Functions: $f(x) = \frac{P(x)}{Q(x)}$, where $P$ and $Q$ are polynomials
• Algebraic Functions: Functions constructed using algebraic operations
• Trigonometric Functions: $\sin x, \cos x, \tan x, \csc x, \sec x, \cot x$
• Exponential Functions: $f(x) = a^x$
• Logarithmic Functions: $f(x) = \log_a x$

Transformations

Given the graph of $y = f(x)$:

• $y = f(x) + c$: Shifts the graph upward by $c$ units.
• $y = f(x) - c$: Shifts the graph downward by $c$ units.
• $y = f(x - c)$: Shifts the graph to the right by $c$ units.
• $y = f(x + c)$: Shifts the graph to the left by $c$ units.
• $y = -f(x)$: Reflects the graph about the x-axis.
• $y = f(-x)$: Reflects the graph about the y-axis.
• $y = c f(x)$: Vertically stretches (if $c > 1$) or compresses (if $0 < c < 1$) the graph.
• $y = f(cx)$: Horizontally stretches or compresses the graph.

Combining Functions

Given two functions $f$ and $g$:

• $(f+g)(x) = f(x) + g(x)$
• $(f-g)(x) = f(x) - g(x)$
• $(fg)(x) = f(x)g(x)$
• $(f/g)(x) = f(x)/g(x)$
• $(f \circ g)(x) = f(g(x))$ is the composite function