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Calculus: Functions of One Variable

1. Functions and Its Representations

Definition:

A function is a relation between a set of inputs (domain) and a set of possible outputs (range) such that each input is related to exactly one output.

Representations of Functions:

  1. Algebraic Representation: A function can be expressed as a formula, e.g., \( f(x) = x^2 + 3x + 5 \).

  2. Graphical Representation: A graph of a function shows the relationship between \( x \) (input) and \( f(x) \) (output).

  3. Tabular Representation: A table can list specific \( x \)-values and their corresponding \( f(x) \)-values.

  4. Verbal Description: A function can also be described in words, e.g., "A function that squares a number and adds 2."

Example:

Algebraic Representation:

\[ f(x) = x^2 - 4 \]

Graphical Representation:

Plot the points for \( x = -2, -1, 0, 1, 2 \):

\( x \) \( f(x) \)
-2 0
-1 -3
0 -4
1 -3
2 0

Verbal Representation:

"A function that takes a number, squares it, and subtracts 4."

2. Linear Mathematical Model

Definition:

A linear mathematical model represents a relationship between variables that can be expressed in the form \( y = mx + b \), where:

  • \( m \): Slope of the line
  • \( b \): Y-intercept

Key Features:

  • The graph is a straight line.
  • Slope \( m \): Describes the rate of change.
  • Y-intercept \( b \): The value of \( y \) when \( x = 0 \).

Example:

Problem:

Create a linear model for a car traveling at a constant speed of 60 km/h.

Solution:

Let \( x \) represent time (in hours) and \( y \) represent the distance traveled (in km).

The relationship can be modeled as:

\[ y = 60x \]

Graphical Representation:

If \( x = 1, 2, 3 \):

\( x \) (Time in hours) \( y \) (Distance in km)
1 60
2 120
3 180

3. Combinations of Functions

Definition:

Functions can be combined using arithmetic operations (addition, subtraction, multiplication, and division) or composition.

Arithmetic Combinations:

  1. Addition: \( (f + g)(x) = f(x) + g(x) \)
  2. Subtraction: \( (f - g)(x) = f(x) - g(x) \)
  3. Multiplication: \( (f \cdot g)(x) = f(x) \cdot g(x) \)
  4. Division: \( \left( \frac{f}{g} \right)(x) = \frac{f(x)}{g(x)}, g(x) \neq 0 \)

Composition of Functions:

The composition \( (f \circ g)(x) \) is defined as:

\[ (f \circ g)(x) = f(g(x)) \]

Example:

Given Functions:

\( f(x) = 2x + 3 \) and \( g(x) = x^2 \).

Combinations:

  1. Addition: \( (f + g)(x) = (2x + 3) + (x^2) = x^2 + 2x + 3 \)
  2. Subtraction: \( (f - g)(x) = (2x + 3) - (x^2) = -x^2 + 2x + 3 \)
  3. Multiplication: \( (f \cdot g)(x) = (2x + 3)(x^2) = 2x^3 + 3x^2 \)
  4. Composition: \( (f \circ g)(x) = f(g(x)) = f(x^2) = 2(x^2) + 3 = 2x^2 + 3 \)

4. Rational, Trigonometric, Exponential, and Logarithmic Functions

Rational Functions:

A rational function is the ratio of two polynomials:

\[ R(x) = \frac{P(x)}{Q(x)} \]

where \( Q(x) \neq 0 \).

Example:

\[ R(x) = \frac{x^2 - 1}{x - 2} \]

Key Points:

  • The domain excludes values that make \( Q(x) = 0 \).

Trigonometric Functions:

These include sine, cosine, tangent, etc., and are periodic in nature.

Example:

\( f(x) = \sin(x) \) - Domain: \( (-\infty, \infty) \) - Range: \( [-1, 1] \)

Key Properties:

  • Periodicity: \( \sin(x + 2\pi) = \sin(x) \)
  • Symmetry: \( \sin(-x) = -\sin(x) \)

Exponential Functions:

An exponential function has the form:

\[ f(x) = a \cdot b^x \]

where \( b > 0 \) and \( b \neq 1 \).

Example:

\( f(x) = 2^x \)

Key Points:

  • Growth: If \( b > 1 \).
  • Decay: If \( 0 < b < 1 \).

Logarithmic Functions:

A logarithmic function is the inverse of an exponential function:

\[ f(x) = \log_b(x) \]

where \( b > 0 \) and \( b \neq 1 \).

Example:

\( f(x) = \log_2(x) \)

Key Points:

  • Domain: \( (0, \infty) \)
  • Range: \( (-\infty, \infty) \)