Chapter 1 Functions And Their Graphs

Functions and their graphs form the bedrock of calculus and higher-level mathematics. Understanding their properties, behavior, and representations is crucial for success in these fields. This chapter serves as a foundation, exploring the fundamental concepts of functions, their graphical representation, and various transformations that can be applied to them.

What is a Function?

At its core, a function is a rule that assigns a unique output value to each input value. Think of it as a machine: you put something in (the input), and it spits something else out (the output) according to a specific set of instructions.

Mathematically, we define a function f from a set A to a set B as a rule that assigns to each element x in A exactly one element y in B. We write this as f(x) = y, where:

• x is the input, also known as the independent variable or the argument of the function.
• y is the output, also known as the dependent variable or the value of the function at x.
• A is the domain of the function, representing the set of all possible input values.
• B is the codomain of the function, representing the set of all possible output values.
• The range of the function is the set of all actual output values, which is a subset of the codomain.

Key characteristics of a function:

• Uniqueness: For each input x, there can only be one output f(x). A function cannot produce multiple outputs for the same input.
• Defined for all inputs in the domain: The function must be defined for every element in its domain. You can't have an input value for which the function doesn't provide an output.

Different Ways to Represent a Function

Functions can be represented in several ways:

• Equation: This is the most common representation, using a mathematical formula to define the relationship between the input and output (e.g., f(x) = x² + 1).
• Table of Values: A table lists specific input values and their corresponding output values. This is useful for representing functions where the relationship isn't easily expressed as an equation.
• Graph: A visual representation of the function on a coordinate plane, where the x-axis represents the input and the y-axis represents the output.
• Verbal Description: Describing the function in words, explaining the rule that connects inputs to outputs.
• Mapping Diagram: A diagram that visually connects elements from the domain to elements in the codomain using arrows.

Domain and Range: Finding the Limits

Determining the domain and range of a function is crucial for understanding its behavior and limitations.

• Domain: The domain is the set of all possible x-values that can be plugged into the function without causing any undefined results. Common restrictions that limit the domain include:

  • Division by zero: The denominator of a fraction cannot be zero.
  • Square roots of negative numbers: You can't take the square root (or any even root) of a negative number within the real number system.
  • Logarithms of non-positive numbers: You can only take the logarithm of positive numbers.

• Range: The range is the set of all possible y-values that the function can produce. Finding the range can be more challenging than finding the domain, as it often requires analyzing the function's behavior and identifying its maximum and minimum values.

Examples:

1. f(x) = 1/x

   • Domain: All real numbers except x = 0 (because division by zero is undefined). In interval notation: (-∞, 0) ∪ (0, ∞)
   • Range: All real numbers except y = 0. In interval notation: (-∞, 0) ∪ (0, ∞)

2. g(x) = √x

   • Domain: All real numbers greater than or equal to 0 (because you can't take the square root of a negative number). In interval notation: [0, ∞)
   • Range: All real numbers greater than or equal to 0. In interval notation: [0, ∞)

3. h(x) = x²

   • Domain: All real numbers. In interval notation: (-∞, ∞)
   • Range: All real numbers greater than or equal to 0. In interval notation: [0, ∞)

Graphing Functions: Visualizing the Relationship

The graph of a function is a powerful tool for visualizing its behavior and properties. It provides a visual representation of the relationship between the input and output values.

Creating a Graph:

1. Choose a set of x-values: Select a representative set of input values within the function's domain.
2. Calculate the corresponding y-values: For each x-value, calculate the corresponding y-value using the function's equation: y = f(x).
3. Plot the points: Plot each (x, y) pair as a point on the coordinate plane.
4. Connect the points: Connect the points with a smooth curve or line to create the graph of the function. The nature of the curve depends on the type of function.

Key features to observe on a graph:

• x-intercepts: Points where the graph intersects the x-axis (where y = 0). These are also called roots or zeros of the function.
• y-intercept: Point where the graph intersects the y-axis (where x = 0).
• Increasing/Decreasing intervals: Intervals where the function's value is increasing or decreasing as x increases.
• Maximum and Minimum points: Points where the function reaches its highest or lowest value within a specific interval (local maxima and minima) or over the entire domain (absolute maximum and minimum).
• Symmetry: Whether the graph is symmetric about the y-axis (even function), the origin (odd function), or neither.
• Asymptotes: Lines that the graph approaches but never touches, indicating the function's behavior as x approaches infinity or a specific value.

Common Types of Functions and Their Graphs

Understanding the graphs of common types of functions is essential for recognizing and analyzing more complex functions. Here are a few examples:

• Linear Function: f(x) = mx + b, where m is the slope and b is the y-intercept. The graph is a straight line.
• Quadratic Function: f(x) = ax² + bx + c, where a, b, and c are constants. The graph is a parabola.
• Cubic Function: f(x) = ax³ + bx² + cx + d, where a, b, c, and d are constants. The graph has a more complex curve with potential inflection points.
• Square Root Function: f(x) = √x. The graph starts at (0, 0) and increases gradually, curving to the right.
• Absolute Value Function: f(x) = |x|. The graph is V-shaped, with the vertex at (0, 0).
• Rational Function: f(x) = p(x) / q(x), where p(x) and q(x) are polynomials. The graph can have vertical and horizontal asymptotes.
• Exponential Function: f(x) = aˣ, where a is a constant and a > 0. The graph increases rapidly as x increases (if a > 1) or decreases rapidly (if 0 < a < 1).
• Logarithmic Function: f(x) = logₐ(x), where a is a constant and a > 0, a ≠ 1. The graph is the inverse of the exponential function.
• Trigonometric Functions: f(x) = sin(x), f(x) = cos(x), f(x) = tan(x), etc. The graphs are periodic, repeating their pattern over regular intervals.

Transformations of Functions: Shifting, Stretching, and Reflecting

Understanding how to transform functions is a powerful tool for creating new functions from existing ones and for analyzing the relationships between different functions. There are several common types of transformations:

• Vertical Shifts:

  • f(x) + c: Shifts the graph of f(x) upward by c units.
  • f(x) - c: Shifts the graph of f(x) downward by c units.

• Horizontal Shifts:

  • f(x - c): Shifts the graph of f(x) to the right by c units.
  • f(x + c): Shifts the graph of f(x) to the left by c units.

• Vertical Stretches and Compressions:

  • c * f(x), where c > 1: Stretches the graph of f(x) vertically by a factor of c.
  • c * f(x), where 0 < c < 1: Compresses the graph of f(x) vertically by a factor of c.

• Horizontal Stretches and Compressions:

  • f(c * x), where c > 1: Compresses the graph of f(x) horizontally by a factor of c.
  • f(c * x), where 0 < c < 1: Stretches the graph of f(x) horizontally by a factor of c.

• Reflections:

  • -f(x): Reflects the graph of f(x) across the x-axis.
  • f(-x): Reflects the graph of f(x) across the y-axis.

Order of Transformations:

When applying multiple transformations, the order matters. A general guideline is to follow this order:

1. Horizontal shifts
2. Horizontal stretches/compressions and reflections about the y-axis
3. Vertical stretches/compressions and reflections about the x-axis
4. Vertical shifts

Example:

Consider the function f(x) = x². Let's apply the following transformations:

1. Shift left by 2 units: f(x + 2) = (x + 2)²
2. Reflect across the x-axis: -f(x + 2) = -(x + 2)²
3. Shift up by 3 units: -f(x + 2) + 3 = -(x + 2)² + 3

The resulting function is g(x) = -(x + 2)² + 3. The graph of g(x) is a parabola that opens downwards, has its vertex at (-2, 3), and is a transformation of the original parabola f(x) = x².

Combining Functions: Arithmetic Operations and Composition

Functions can be combined using arithmetic operations and composition to create new functions.

Arithmetic Operations

Functions can be added, subtracted, multiplied, and divided, just like numbers. Given two functions f(x) and g(x), we can define the following operations:

• Sum: (f + g)(x) = f(x) + g(x)
• Difference: (f - g)(x) = f(x) - g(x)
• Product: (f * g)(x) = f(x) * g(x)
• Quotient: (f / g)(x) = f(x) / g(x), where g(x) ≠ 0

The domain of the resulting function is the intersection of the domains of f(x) and g(x), with the additional restriction that g(x) ≠ 0 in the case of the quotient.

Example:

Let f(x) = x² and g(x) = x + 1. Then:

• (f + g)(x) = x² + x + 1
• (f - g)(x) = x² - x - 1
• (f * g)(x) = x²(x + 1) = x³ + x²
• (f / g)(x) = x² / (x + 1), where x ≠ -1

Composition of Functions

The composition of two functions f and g, denoted by f ∘ g, is defined as (f ∘ g)(x) = f(g(x)). This means that you first apply the function g to the input x, and then apply the function f to the result g(x).

The domain of f ∘ g is the set of all x in the domain of g such that g(x) is in the domain of f. In simpler terms, the output of g must be a valid input for f.

Example:

Let f(x) = √x and g(x) = x + 2. Then:

• (f ∘ g)(x) = f(g(x)) = f(x + 2) = √(x + 2)
• (g ∘ f)(x) = g(f(x)) = g(√x) = √x + 2

Notice that f ∘ g and g ∘ f are generally not the same.

• Domain of f ∘ g: x + 2 ≥ 0 => x ≥ -2. In interval notation: [-2, ∞)
• Domain of g ∘ f: x ≥ 0. In interval notation: [0, ∞) (because you can only take the square root of non-negative numbers)

Inverse Functions: Undoing the Operation

An inverse function "undoes" the action of the original function. If f(x) = y, then the inverse function, denoted by f⁻¹(y), satisfies f⁻¹(y) = x.

Conditions for the Existence of an Inverse Function:

A function f has an inverse function if and only if it is one-to-one. A function is one-to-one if each output value corresponds to exactly one input value. Graphically, this means that the function passes the horizontal line test: no horizontal line intersects the graph of the function more than once.

Finding the Inverse Function:

1. Replace f(x) with y: Write the equation as y = f(x).
2. Swap x and y: Interchange x and y to get x = f(y).
3. Solve for y: Solve the equation for y in terms of x. This gives you y = f⁻¹(x).
4. Replace y with f⁻¹(x): Write the inverse function as f⁻¹(x) = ...

Example:

Find the inverse of f(x) = 2x + 3.

1. y = 2x + 3
2. x = 2y + 3
3. x - 3 = 2y
4. y = (x - 3) / 2
5. f⁻¹(x) = (x - 3) / 2

Graphs of Inverse Functions:

The graph of f⁻¹(x) is the reflection of the graph of f(x) across the line y = x.

Important Properties:

• f(f⁻¹(x)) = x for all x in the domain of f⁻¹.
• f⁻¹(f(x)) = x for all x in the domain of f.

Symmetry: Even and Odd Functions

Symmetry is an important property of functions that can simplify their analysis and graphing. There are two main types of symmetry:

• Even Function: A function f(x) is even if f(-x) = f(x) for all x in its domain. The graph of an even function is symmetric about the y-axis. Examples: f(x) = x², f(x) = cos(x), f(x) = |x|.

• Odd Function: A function f(x) is odd if f(-x) = -f(x) for all x in its domain. The graph of an odd function is symmetric about the origin. Examples: f(x) = x³, f(x) = sin(x), f(x) = x.

Testing for Even and Odd Functions:

1. Find f(-x): Replace x with -x in the function's equation.
2. Compare f(-x) with f(x):
   • If f(-x) = f(x), the function is even.
   • If f(-x) = -f(x), the function is odd.
   • If neither of these conditions holds, the function is neither even nor odd.

Example:

• f(x) = x⁴ + 2x²:

  • f(-x) = (-x)⁴ + 2(-x)² = x⁴ + 2x² = f(x). Therefore, f(x) is even.

• g(x) = x³ - x:

  • g(-x) = (-x)³ - (-x) = -x³ + x = -(x³ - x) = -g(x). Therefore, g(x) is odd.

• h(x) = x² + x:

  • h(-x) = (-x)² + (-x) = x² - x. This is neither equal to h(x) nor -h(x). Therefore, h(x) is neither even nor odd.

Piecewise-Defined Functions: Functions with Multiple Rules

A piecewise-defined function is a function defined by multiple sub-functions, where each sub-function applies to a specific interval of the domain.

Example:

f(x) = {
    x²  if x < 0
    x   if 0 ≤ x ≤ 1
    1   if x > 1
}

This function is defined as x² for all x less than 0, as x for all x between 0 and 1 (inclusive), and as 1 for all x greater than 1.

Graphing Piecewise-Defined Functions:

To graph a piecewise-defined function, graph each sub-function over its corresponding interval. Be careful to use open circles (o) at the endpoints of intervals where the function is not defined at that point, and closed circles (•) where the function is defined.

Example:

For the function above:

• For x < 0, graph the parabola y = x². Use an open circle at (0, 0).
• For 0 ≤ x ≤ 1, graph the line y = x. Use a closed circle at (0, 0) and (1, 1).
• For x > 1, graph the horizontal line y = 1. Use an open circle at (1, 1).

Frequently Asked Questions (FAQ)

Q: What's the difference between a relation and a function?

A: A relation is any set of ordered pairs. A function is a special type of relation where each input has exactly one output.

Q: How do I determine the domain of a function?

A: Look for restrictions such as division by zero, square roots of negative numbers, and logarithms of non-positive numbers.

Q: How do I determine the range of a function?

A: Analyze the function's behavior, identify its maximum and minimum values, and consider any asymptotes.

Q: What is the vertical line test?

A: The vertical line test is a visual test to determine if a graph represents a function. If any vertical line intersects the graph more than once, the graph does not represent a function.

Q: What is the horizontal line test?

A: The horizontal line test is a visual test to determine if a function is one-to-one (and therefore has an inverse). If any horizontal line intersects the graph more than once, the function is not one-to-one.

Q: How do I find the inverse of a function?

A: Swap x and y in the function's equation and solve for y.

Q: What is the difference between an even and an odd function?

A: An even function is symmetric about the y-axis (f(-x) = f(x)), while an odd function is symmetric about the origin (f(-x) = -f(x)).

Q: Why are functions important?

A: Functions are fundamental building blocks in mathematics and are used to model relationships between quantities in various fields, including science, engineering, economics, and computer science.

Conclusion

This chapter has provided a comprehensive overview of functions and their graphs, covering key concepts such as domain, range, transformations, combinations, inverse functions, and symmetry. A solid understanding of these concepts is crucial for success in calculus and other advanced mathematics courses. By mastering the techniques presented here, you will be well-equipped to analyze, manipulate, and apply functions in a variety of contexts. Remember to practice regularly and work through examples to solidify your understanding. Good luck!