Functions & Graphs

TL;DR

Functions take an input, do something to it, and give you an output. Graphs are visual maps of these input-output relationships. Understanding both helps you predict behavior and visualize data.

1. The Mental Model

Think of a function as a machine: you put something in, the machine processes it, and something comes out. A graph is like drawing a picture of what that machine does for every input you can imagine.

2. The Core Material

What is a Function?

A function assigns exactly one output for each input. It's a rule that connects an input value (often called $x$) to an output value (often called $y$ or $f(x)$).

For example, $f(x) = x + 2$ is a function. If you input 3, the output is $3 + 2 = 5$. If you input -1, the output is $-1 + 2 = 1$. Each input only has one possible output. If an input could give two different outputs, it wouldn't be a function.

Domain and Range

The domain is the set of all possible input values for a function.
The range is the set of all possible output values that the function can produce.

For $f(x) = x + 2$, you can put any real number in, so the domain is all real numbers. Since the outputs can also be any real number, the range is all real numbers.

Consider $f(x) = \sqrt{x}$. You can't take the square root of a negative number and get a real number. So, the domain is $x \ge 0$. The outputs will always be 0 or positive, so the range is $y \ge 0$.

How to Graph a Function

A graph is a visual representation of a function, typically on a coordinate plane. The horizontal axis (x-axis) represents the input values, and the vertical axis (y-axis) represents the output values. Each point $(x, y)$ on the graph shows an input $x$ and its corresponding output $y$.

To graph a simple function:
1. Pick some input values (x-values): Choose a few numbers, both positive and negative, and zero.
2. Calculate the output values (y-values): Use the function's rule to find the corresponding output for each input.
3. Plot the points: For each pair $(x, y)$, mark that spot on the coordinate plane.
4. Connect the dots: Draw a line or curve through the points, assuming the domain allows for values in between your chosen points.

Types of Functions (and their graphs)

You'll encounter many types, but here are a few basic ones:

• Linear Functions: $f(x) = mx + b$. Their graphs are straight lines. 'm' is the slope (steepness), and 'b' is the y-intercept (where it crosses the y-axis).
  • Example: $f(x) = 2x + 1$
• Quadratic Functions: $f(x) = ax^2 + bx + c$. Their graphs are parabolas (U-shaped curves).
  • Example: $f(x) = x^2$
• Absolute Value Functions: $f(x) = |x|$. Their graphs are V-shaped.
  • Example: $f(x) = |x - 2|$

Vertical Line Test

A quick way to check if a graph represents a function is the Vertical Line Test. If you can draw any vertical line that crosses the graph at more than one point, then it's not a function. This is because a single input (x-value) would have multiple outputs (y-values), violating the definition of a function.

3. Worked Example

Let's graph the function $f(x) = -x + 3$.

1. Choose inputs (x-values): Let's use -2, 0, and 3.
2. Calculate outputs (y-values):
   • For $x = -2$: $f(-2) = -(-2) + 3 = 2 + 3 = 5$. So, the point is $(-2, 5)$.
   • For $x = 0$: $f(0) = -(0) + 3 = 0 + 3 = 3$. So, the point is $(0, 3)$.
   • For $x = 3$: $f(3) = -(3) + 3 = -3 + 3 = 0$. So, the point is $(3, 0)$.
3. Plot the points: Imagine an x-y plane.
   • Mark the point 2 units left and 5 units up from the origin.
   • Mark the point on the y-axis at 3.
   • Mark the point on the x-axis at 3.
4. Connect the dots: Draw a straight line through $(-2, 5)$, $(0, 3)$, and $(3, 0)$. This is the graph of $f(x) = -x + 3$.

4. Key Takeaways

• A function is a rule where every input has exactly one output.
• The domain is all possible inputs, and the range is all possible outputs.
• Graphs visualize input-output pairs $(x, y)$ on a coordinate plane.
• The Vertical Line Test helps identify if a graph represents a function.
• Linear functions graph as straight lines, quadratics as parabolas, absolute value as V-shapes.
• Plotting a few calculated points helps you sketch a graph.
• Use $f(x)$ notation as a convenient way to represent the output of a function for a given input $x$.

Common Mistakes to Avoid

• Assuming every equation is a function; check the Vertical Line Test!
• Confusing domain (x-values) with range (y-values).
• Connecting points with a curve or line when the domain is only-specific integers (e.g., "number of apples").
• Forgetting that negative signs outside parentheses ($ -x^2 $) are different from inside ($ (-x)^2 $).

5. Now Try It

Graph the function $g(x) = x^2 - 1$. Pick integer input values from -2 to 2 (e.g., -2, -1, 0, 1, 2), calculate their corresponding output values, plot these points, and then draw the curve that connects them. What shape do you get?