1.3 Functions and Their Graphs

A function is a mathematical relation between a set of inputs (called the domain) and a set of possible outputs (called the range), where each input is related to exactly one output. Understanding functions and their graphical representations is crucial in calculus, as many of the key concepts in calculus (such as limits, derivatives, and integrals) are centered around functions.

1.3.1 Definition of a Function

Formally, a function \( f \) from a set \( A \) to a set \( B \) is a rule that assigns each element \( x \) in \( A \) to exactly one element \( y \) in \( B \). This relationship is typically written as:
\[
y = f(x)
\]
Here, \( x \) is the independent variable, and \( y \) (or \( f(x) \)) is the dependent variable.

Example:
Let \( f(x) = 2x + 3 \). For every value of \( x \), there is a corresponding unique value of \( f(x) \). For example:

• \( f(1) = 2(1) + 3 = 5 \)
• \( f(-2) = 2(-2) + 3 = -1 \)

1.3.2 Domain and Range of a Function

The domain of a function is the set of all possible input values (values of \( x \)) that the function can accept. The range is the set of all possible output values (values of \( f(x) \)).

• Domain: The set of \( x \)-values for which the function is defined.
• Range: The set of \( y \)-values that the function can output.

Example:
For the function \( f(x) = \frac{1}{x} \), the domain is \( x \neq 0 \), since division by zero is undefined. The range is \( f(x) \in \mathbb{R} \setminus {0} \).

1.3.3 Types of Functions

1. Linear Functions:
   A linear function has the form:
   \[
   f(x) = mx + b
   \]
   where \( m \) is the slope and \( b \) is the \( y \)-intercept. The graph of a linear function is a straight line. Example: \( f(x) = 2x + 3 \) is a linear function with slope 2 and \( y \)-intercept 3.
2. Quadratic Functions:
   A quadratic function has the form:
   \[
   f(x) = ax^2 + bx + c
   \]
   where \( a \neq 0 \). The graph of a quadratic function is a parabola. Example: \( f(x) = x^2 - 4x + 3 \). This parabola opens upwards (since \( a > 0 \)) and has a vertex and axis of symmetry.
3. Polynomial Functions:
   A polynomial function is a sum of terms, each consisting of a variable raised to a non-negative integer power multiplied by a coefficient. A general polynomial function has the form:
   \[
   f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0
   \]
   where \( a_n \neq 0 \) and \( n \) is a non-negative integer. Example: \( f(x) = 3x^3 - 2x^2 + x - 5 \)
4. Rational Functions:
   A rational function is the ratio of two polynomial functions. It has the form:
   \[
   f(x) = \frac{P(x)}{Q(x)}
   \]
   where \( P(x) \) and \( Q(x) \) are polynomials, and \( Q(x) \neq 0 \). Example: \( f(x) = \frac{x^2 - 1}{x - 2} \)
5. Exponential Functions:
   An exponential function has the form:
   \[
   f(x) = a^x
   \]
   where \( a > 0 \) and \( a \neq 1 \). The graph of an exponential function grows rapidly (if \( a > 1 \)) or decays rapidly (if \( 0 < a < 1 \)). Example: \( f(x) = 2^x \)
6. Logarithmic Functions:
   A logarithmic function is the inverse of an exponential function. It has the form:
   \[
   f(x) = \log_a(x)
   \]
   where \( a > 0 \) and \( a \neq 1 \). Example: \( f(x) = \log_2(x) \)
7. Trigonometric Functions:
   Trigonometric functions involve the angles of a right triangle. The most common are:
   \[
   \sin(x), \quad \cos(x), \quad \tan(x)
   \]
   These functions are periodic, meaning they repeat their values at regular intervals. Example: \( f(x) = \sin(x) \)

1.3.4 Graphing Functions

The graph of a function is a visual representation of the relationship between the input and output values. To graph a function, plot points corresponding to pairs \( (x, f(x)) \), and then connect the points smoothly, if applicable.

Example:
Consider the function \( f(x) = x^2 - 4x + 3 \). To graph it:

1. Choose values for \( x \), such as \( x = -1, 0, 1, 2, 3, 4 \).
2. Calculate the corresponding \( f(x) \)-values.
   \[
   f(-1) = (-1)^2 - 4(-1) + 3 = 8
   \]
   \[
   f(0) = 0^2 - 4(0) + 3 = 3
   \]
   \[
   f(1) = 1^2 - 4(1) + 3 = 0
   \]
   \[
   f(2) = 2^2 - 4(2) + 3 = -1
   \]
   \[
   f(3) = 3^2 - 4(3) + 3 = 0
   \]
   \[
   f(4) = 4^2 - 4(4) + 3 = 3
   \]
3. Plot the points \( (-1, 8), (0, 3), (1, 0), (2, -1), (3, 0), (4, 3) \) and connect them to form the parabola.

1.3.5 Symmetry of Graphs

A function's graph can exhibit symmetry, which simplifies graphing and analyzing the function.

1. Even Functions:
   A function is even if:
   \[
   f(-x) = f(x)
   \]
   for all \( x \) in the domain. The graph of an even function is symmetric with respect to the \( y \)-axis.Example: \( f(x) = x^2 \)
2. Odd Functions:
   A function is odd if:
   \[
   f(-x) = -f(x)
   \]
   for all \( x \) in the domain. The graph of an odd function is symmetric with respect to the origin.Example: \( f(x) = x^3 \)

1.3.6 Transformations of Functions

Graph transformations allow you to modify the shape and position of a function's graph.

1. Vertical Shifts:
   \[
   f(x) + c
   \]
   shifts the graph up by \( c \) units if \( c > 0 \), and down by \( c \) units if \( c < 0 \).
2. Horizontal Shifts:
   \[
   f(x - c)
   \]
   shifts the graph to the right by \( c \) units if \( c > 0 \), and to the left by \( c < 0 \).
3. Vertical Stretch/Compression:
   \[
   a \cdot f(x)
   \]
   stretches the graph vertically if \( |a| > 1 \) and compresses it if \( 0 < |a| < 1 \).
4. Horizontal Stretch/Compression:
   \[
   f(kx)
   \]
   compresses the graph horizontally if \( |k| > 1 \), and stretches it horizontally if \( 0 < |k| < 1 \).
5. Reflections:
   • \( -f(x) \): Reflects the graph across the \( x \)-axis.
   • \( f(-x) \): Reflects the graph across the \( y \)-axis.

Example:
Given \( f(x) = x^2 \):

• \( f(x) + 3 \) shifts the parabola upward by 3 units.
• \( f(x - 2) \) shifts the parabola to the right by

2 units.

• \( 2f(x) \) vertically stretches the parabola.

1.3.7 Combining Functions

You can combine functions in various ways to create new functions:

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:
   \[
   \left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} \quad \text{where} \quad g(x) \neq 0
   \]
5. Composition:
   \[
   (f \circ g)(x) = f(g(x))
   \]
   This means you apply the function ( g ) first, then apply the function ( f ) to the result.

Example:
Given \( f(x) = x^2 \) and \( g(x) = 3x + 1 \), the composition \( (f \circ g)(x) \) is:
\[
(f \circ g)(x) = f(g(x)) = f(3x + 1) = (3x + 1)^2 = 9x^2 + 6x + 1
\]

This concludes the review of functions and their graphs. Mastering these concepts is essential for understanding more advanced topics in calculus, such as limits, derivatives, and integrals.