Chapter 1: Functions – Foundations for Calculus

Chapter 1: Functions

1.1 Review of Functions

Understanding functions is essential for calculus. This section introduces the concept of functions, their representations, and foundational properties.

• Relation: Any connection between pairs of numbers. An element from the input can have more than one element in the output.

• Function: A rule that assigns to each value x in a set D a unique value denoted f(x). The set D is the domain of the function. The range is the set of all values of f(x) produced as x varies over the domain.

• Independent variable: Variable associated with the domain (input).

• Dependent variable: Variable associated with the range (output).

• Graph of a function: The set of all points (x, f(x)).

Representations of Functions

• Numerical: Table of values

• Graphical: Graph

• Symbolic: Formula

Vertical Line Test

A graph represents a function if and only if every vertical line intersects the graph at most once. If a vertical line crosses more than once, the graph does not represent a function.

Examples

• Graph the following functions and identify their domains and ranges:

• For y = x^2 + 1: Domain: $(-\infty, \infty)$ Range: $[1, \infty)$

• For z = \sqrt{4 - t^2}: Domain: $[-2, 2]$ Range: $[0, 2]$

• For w = \frac{1}{u-1}: Domain: $\{u \mid u \neq 1\}$ Range: $(-\infty, 0) \cup (0, \infty)$

Function Operations

• Sum/Difference: $(f \pm g)(x) = f(x) \pm g(x)$

• Product: $(fg)(x) = f(x)g(x)$

• Quotient: $\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}$, $g(x) \neq 0$

Composite Functions

Given two functions $f$ and $g$, the composite function $f \circ g$ is defined by:

$(f \circ g)(x) = f(g(x))$

• Evaluated in two steps: $u = g(x)$, then $y = f(u)$.

• The domain of $f \circ g$ consists of all $x$ in the domain of $g$ such that $g(x)$ is in the domain of $f$.

• Commutative property does not apply: $f \circ g \neq g \circ f$ in general.

Secant Line and Difference Quotient

The secant line is a line through two points on a curve. Its slope is called the difference quotient:

• First form (points $x$ and $x+h$): $m_{sec} = \frac{f(x+h) - f(x)}{h}$

• Second form (points $a$ and $x$): $m_{sec} = \frac{f(x) - f(a)}{x - a}$

Application: Velocity of a Moving Object

• The position $x$ (in meters) of an object as a function of time $t$ (in seconds) can be used to calculate average velocity between two points:

• Average velocity between $t_1$ and $t_2$: $\text{Average velocity} = \frac{x(t_2) - x(t_1)}{t_2 - t_1}$

Symmetry in Graphs

• Symmetry about the y-axis: $f(x) = f(-x)$ (Even function)

• Symmetry about the x-axis: Not a function (fails vertical line test)

• Symmetry about the origin: $f(-x) = -f(x)$ (Odd function)

Even and Odd Functions

• Even function: $f(-x) = f(x)$ for all $x$ in the domain. Graph is symmetric about the y-axis. Example: $f(x) = x^2$.

• Odd function: $f(-x) = -f(x)$ for all $x$ in the domain. Graph is symmetric about the origin. Example: $f(x) = x^3$.

• Polynomials with only even powers are even; with only odd powers are odd; with both, neither.

1.2 Representing Functions

Functions can be classified into several types, each with unique properties and applications.

Types of Functions

• Polynomial functions: $f(x) = a_n x^n + \ldots + a_1 x + a_0$ - Degree: highest power of $x$ - Leading coefficient: $a_n$ - $n$ is a nonnegative integer

• Rational functions: $f(x) = \frac{p(x)}{q(x)}$, where $p(x)$ and $q(x)$ are polynomials and $q(x) \neq 0$

• Algebraic functions: Functions involving algebraic operations, not expressible as polynomials or rational functions (e.g., $f(x) = \sqrt{x}$)

• Exponential and logarithmic functions: $f(x) = a^x$, $f(x) = \log_a x$

• Trigonometric functions: $f(x) = \sin x$, $f(x) = \cos x$, etc.

Roots of Functions

• Roots (zeros): Values of $x$ for which $f(x) = 0$.

• A polynomial of degree $n$ can have up to $n$ roots.

• Quadratic example: $g(x) = 3x^2$ has one root at $x = 0$; $g(x) = 3x^2 + 1$ has no real roots (imaginary roots).

Linear Functions

• Form: $f(x) = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

• Parallel lines have the same slope but different y-intercepts.

• Slope between points $(x_1, y_1)$ and $(x_2, y_2)$: $m = \frac{y_2 - y_1}{x_2 - x_1}$

• Equation of a line with slope $m$ through point $(x_1, y_1)$: $y - y_1 = m(x - x_1)$

Piecewise Functions

• Defined by different expressions on different intervals of the domain.

• Example: $g(x) = \begin{cases} x, & x < 2 \\ 3, & x = 2 \\ \frac{x}{2} + 5, & x > 2 \end{cases}$

Rational Functions

• Form: $f(x) = \frac{p(x)}{q(x)}$, $q(x) \neq 0$

• Domain: All $x$ such that $q(x) \neq 0$

A Library of Functions

Type	Example
Linear function	$f(x) = x$
Quadratic function	$f(x) = x^2$
Constant function	$f(x) = c$
Absolute-value function	$f(x) = |x|$
Square-root function	$f(x) = \sqrt{x}$
Exponential function	$f(x) = a^x$
Cube root function	$f(x) = \sqrt[3]{x}$
Rational function	$f(x) = \frac{1}{x}$
Logarithmic function	$f(x) = \log_a x$

Additional info: This summary covers the foundational concepts of functions, their types, properties, and representations, which are essential for further study in calculus.