Chapter 1 Functions And Their Graphs

Functions and their graphs serve as the bedrock of calculus, providing a visual and analytical framework for understanding relationships between variables. Mastering this foundational material is crucial for success in advanced mathematics and its applications in science, engineering, economics, and beyond.

What is a Function?

A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. Think of it as a machine: you put something in (the input), and it spits something else out (the output). The key is that for every input, you always get the same output.

• Input: The value you put into the function, often denoted by x.
• Output: The value the function gives you, often denoted by y or f(x).
• Domain: The set of all possible input values for the function.
• Range: The set of all possible output values for the function.

Ways to Represent a Function

Functions can be represented in several ways:

1. Equation: This is the most common representation, such as f(x) = x^2 + 3 or y = 2x - 1.
2. Table: A table lists specific input values and their corresponding output values.
3. 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.
4. Words: Describing the relationship between input and output in plain language. For example, "The function doubles the input and adds five."
5. Mapping Diagram: A diagram showing how elements from the domain are mapped to elements in the range.

Identifying Functions

How do we know if a relation is actually a function? The most important test is the Vertical Line Test.

• Vertical Line Test: If any vertical line drawn on the graph of a relation intersects the graph more than once, then the relation is not a function. This is because a single x-value would be associated with multiple y-values, violating the definition of a function.

Common Types of Functions

Let's explore some common types of functions you'll encounter:

1. Linear Functions: These functions have the form f(x) = mx + b, where m is the slope and b is the y-intercept. Their graphs are straight lines.
2. Quadratic Functions: These functions have the form f(x) = ax^2 + bx + c, where a, b, and c are constants and a ≠ 0. Their graphs are parabolas.
3. Polynomial Functions: These functions are sums of terms, each of which is a constant multiplied by a power of x (e.g., f(x) = 3x^4 - 2x^2 + x - 7).
4. Rational Functions: These functions are ratios of two polynomials (e.g., f(x) = (x + 1) / (x^2 - 4)).
5. Exponential Functions: These functions have the form f(x) = a^x, where a is a positive constant (e.g., f(x) = 2^x).
6. Logarithmic Functions: These functions are the inverses of exponential functions (e.g., f(x) = log_2(x)).
7. Trigonometric Functions: These functions relate angles of a right triangle to ratios of its sides (e.g., f(x) = sin(x), f(x) = cos(x)).
8. Absolute Value Function: This function, denoted f(x) = |x|, returns the non-negative value of x.
9. Piecewise Functions: These functions are defined by different formulas on different parts of their domain.

Graphing Functions

Graphing functions allows us to visualize their behavior and properties. Here's a breakdown of the process:

1. Understanding the Coordinate Plane

The coordinate plane, also known as the Cartesian plane, is formed by two perpendicular lines:

• x-axis: The horizontal line.
• y-axis: The vertical line.

The point where the axes intersect is called the origin, and its coordinates are (0, 0). Any point on the plane can be located using an ordered pair (x, y), where x represents the horizontal distance from the origin and y represents the vertical distance from the origin.

2. Plotting Points

To plot a point (x, y):

1. Start at the origin.
2. Move x units horizontally (right if x is positive, left if x is negative).
3. Move y units vertically (up if y is positive, down if y is negative).
4. Mark the location with a dot.

3. Graphing by Plotting Points

A basic way to graph a function is to create a table of values, plot the corresponding points, and then connect the points with a smooth curve.

1. Choose x-values: Select a range of x-values that you think will adequately represent the function's behavior.
2. Calculate y-values: For each chosen x-value, calculate the corresponding y-value using the function's equation.
3. Plot the points: Plot each (x, y) pair on the coordinate plane.
4. Connect the points: Draw a smooth curve (or a line, if it's a linear function) through the plotted points.

Example: Graph the function f(x) = x^2 - 2

Plot the points (-2, 2), (-1, -1), (0, -2), (1, -1), and (2, 2) and connect them with a smooth curve to form a parabola.

4. Using Transformations

Understanding transformations of functions can significantly simplify the graphing process. Here are some common 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.

• 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.

• Vertical Stretches and Compressions:

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

• Horizontal Stretches and Compressions:

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

Example: Let's graph g(x) = (x - 2)^2 + 1. We know the basic function f(x) = x^2 is a parabola centered at the origin. g(x) is a transformation of f(x):

1. x - 2: Shifts the parabola 2 units to the right.
2. + 1: Shifts the parabola 1 unit upward.

Therefore, the graph of g(x) is a parabola with its vertex at (2, 1).

5. Using Key Features

Identifying key features of a function can help you graph it more accurately:

• Intercepts:

  • x-intercept: The point(s) where the graph crosses the x-axis (where y = 0). Found by setting f(x) = 0 and solving for x.
  • y-intercept: The point where the graph crosses the y-axis (where x = 0). Found by evaluating f(0).

• Vertex: The highest or lowest point on a parabola. For a quadratic function f(x) = ax^2 + bx + c, the x-coordinate of the vertex is given by x = -b / 2a.

• Asymptotes: Lines that the graph of a function approaches but never touches. Rational functions often have vertical and horizontal asymptotes.

• Symmetry:

  • Even Function: A function is even if f(-x) = f(x). Its graph is symmetric about the y-axis.
  • Odd Function: A function is odd if f(-x) = -f(x). Its graph is symmetric about the origin.

• End Behavior: What happens to the function's output as x approaches positive or negative infinity.

Function Operations

Functions can be combined and manipulated in several ways:

1. Addition: (f + g)(x) = f(x) + g(x)
2. Subtraction: (f - g)(x) = f(x) - g(x)
3. Multiplication: (f * g)(x) = f(x) * g(x)
4. Division: (f / g)(x) = f(x) / g(x), where g(x) ≠ 0
5. Composition: (f ∘ g)(x) = f(g(x)). This means you first evaluate g(x), and then plug that result into f(x). The domain of the composite function is all x in the domain of g such that g(x) is in the domain of f.

Example: Let f(x) = x^2 and g(x) = x + 1.

• (f + g)(x) = x^2 + x + 1
• (f * g)(x) = x^2(x + 1) = x^3 + x^2
• (f ∘ g)(x) = f(g(x)) = f(x + 1) = (x + 1)^2 = x^2 + 2x + 1

Inverse Functions

An inverse function "undoes" the original function. If f(a) = b, then f^(-1)(b) = a. (Note: f^(-1)(x) does not mean 1/f(x).)

• Finding the Inverse:

  1. Replace f(x) with y.
  2. Swap x and y.
  3. Solve for y.
  4. Replace y with f^(-1)(x).

• Horizontal Line Test: A function has an inverse function if and only if no horizontal line intersects its graph more than once. If a function passes the horizontal line test, it is called one-to-one.

• Composition and Inverses: If f and f^(-1) are inverse functions, then f(f^(-1)(x)) = x and f^(-1)(f(x)) = x for all x in their respective domains.

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

To verify: f(f^(-1)(x)) = 2((x-3)/2) + 3 = (x-3) + 3 = x f^(-1)(f(x)) = ((2x+3)-3)/2 = (2x)/2 = x

Building New Functions From Old

Besides the basic function operations, we can build new functions from existing ones using a variety of techniques:

1. Piecewise-Defined Functions

A piecewise-defined function is defined by different expressions on different intervals of its domain.

Example:

f(x) = {
    x^2,   if x < 0
    x + 1, if x >= 0
}

To graph a piecewise function, graph each piece separately over its specified interval. Be careful at the boundaries where the pieces meet.

2. Even and Odd Functions

• Even Functions: 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 with respect to the y-axis. Examples include x^2, x^4, and cos(x).
• Odd Functions: 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 with respect to the origin. Examples include x, x^3, and sin(x).

Knowing whether a function is even or odd can simplify graphing and analysis.

3. Increasing and Decreasing Functions

• Increasing Function: A function f(x) is increasing on an interval if, for any two numbers x1 and x2 in the interval, x1 < x2 implies f(x1) < f(x2). As you move from left to right along the graph, the y-values are going up.
• Decreasing Function: A function f(x) is decreasing on an interval if, for any two numbers x1 and x2 in the interval, x1 < x2 implies f(x1) > f(x2). As you move from left to right along the graph, the y-values are going down.
• Constant Function: A function f(x) is constant on an interval if, for any two numbers x1 and x2 in the interval, f(x1) = f(x2). The graph is a horizontal line.

Identifying intervals where a function is increasing, decreasing, or constant provides valuable information about its behavior.

Applications of Functions and Their Graphs

Functions and their graphs are essential tools in modeling real-world phenomena:

• Physics: Describing motion, forces, and energy.
• Engineering: Designing structures, circuits, and algorithms.
• Economics: Modeling supply, demand, and market trends.
• Biology: Representing population growth, enzyme kinetics, and disease spread.
• Computer Science: Analyzing algorithms, creating graphics, and developing artificial intelligence.

Example: The height of a ball thrown in the air can be modeled by a quadratic function. The graph of this function is a parabola, and its vertex represents the maximum height the ball reaches.

Common Mistakes to Avoid

• Confusing Domain and Range: Remember that the domain is the set of inputs, while the range is the set of outputs.
• Assuming All Relations are Functions: Always check the vertical line test.
• Incorrectly Applying Transformations: Pay close attention to the order of transformations and the signs.
• Forgetting to Check for Restrictions on the Domain: Rational functions have restrictions where the denominator is zero; logarithmic functions require positive arguments; and square root functions require non-negative arguments.
• Misinterpreting Function Notation: f(x) does not mean f times x. It means the value of the function f at the input x.
• Confusing f^(-1)(x) with 1/f(x): The inverse function is not the reciprocal of the function.

Conclusion

Understanding functions and their graphs is a fundamental skill in mathematics. By mastering the concepts of domain, range, transformations, function operations, and inverse functions, you'll be well-equipped to tackle more advanced topics and apply mathematical principles to real-world problems. Practice graphing various types of functions, identifying their key features, and working with transformations to solidify your understanding. Remember that a solid foundation in functions will pave the way for success in calculus and beyond.