Functions and Graphs Fundamentals

TL;DR

Functions are like machines that take an input, process it, and give you exactly one output. Graphs are visual maps of these input-output relationships, letting you see patterns and behaviors at a glance. Understanding them helps you describe and predict real-world situations mathematically.

1. The Mental Model

Imagine a function as a rule that links one thing to another, always consistently. When you graph that rule, you're just drawing a picture of all the possible pairings of those linked things.

2. The Core Material

At its heart, a function is a special kind of relationship where every input has one and only one output. Think of a soda machine: you press 'Coke' (input), and you get a Coke (output) every single time. You'd never press 'Coke' and sometimes get a Sprite or nothing at all.

Function Notation: $f(x)$

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You'll see functions written as $f(x) = \text{something}$.
- $f$ is just the name of the function (you could use $g(x)$, $h(x)$, etc.).
- $x$ is the input variable (what you put into the function).
- $f(x)$ is the output value (what you get out). It's also often called $y$. So, $y = f(x)$.

Example: If $f(x) = 2x + 1$:
- When $x = 3$, $f(3) = 2(3) + 1 = 7$.
- When $x = -1$, $f(-1) = 2(-1) + 1 = -1$.

Domain and Range

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• Domain: All the possible input values ($x$) that you can put into a function.
• Range: All the possible output values ($y$ or $f(x)$) that come out of a function.

For $f(x) = 2x + 1$, you can put any real number into $x$, and you'll always get a real number out. So, both the domain and range are all real numbers. But for something like $g(x) = \sqrt{x}$, you can't take the square root of a negative number in the real number system, so the domain is $x \ge 0$. Consequently, the range will also be $y \ge 0$.

graph TD
    A["Input (x)"] --> B{{"Function Rule f(x)"}};
    B --> C["Output (f(x) or y)"];
    subgraph Function Machine
        B
    end
    subgraph "Possible Values"
        D("Domain") --> A;
        C --> E("Range");
    end

Graphing Functions

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A graph is a picture of all the $(x, y)$ pairs that satisfy a function. You plot points on a coordinate plane, where the horizontal axis is usually $x$ (input) and the vertical axis is $y$ (output).

To graph a function:
1. Pick several $x$ values (inputs).
2. Use the function rule to calculate the corresponding $y$ values (outputs).
3. Plot these $(x, y)$ points on the coordinate plane.
4. Connect the points to show the continuous relationship.

Example: For $f(x) = x^2 - 2$:
- If $x = -2$, $y = (-2)^2 - 2 = 4 - 2 = 2$. Plot $(-2, 2)$.
- If $x = -1$, $y = (-1)^2 - 2 = 1 - 2 = -1$. Plot $(-1, -1)$.
- If $x = 0$, $y = (0)^2 - 2 = 0 - 2 = -2$. Plot $(0, -2)$.
- If $x = 1$, $y = (1)^2 - 2 = 1 - 2 = -1$. Plot $(1, -1)$.
- If $x = 2$, $y = (2)^2 - 2 = 4 - 2 = 2$. Plot $(2, 2)$.
Connecting these points would form a parabola.

The Vertical Line Test

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A quick way to check if a graph represents a function is the Vertical Line Test. If you can draw any vertical line anywhere on the graph that intersects the graph more than once, then it's not a function. This is because multiple intersections mean one $x$-input has more than one $y$-output, which violates the function rule.

3. Worked Example

Let's work with the function $g(x) = 3x - 2$.

1. Find $g(4)$ and $g(-1)$.
- For $g(4)$: Replace $x$ with 4.
$g(4) = 3(4) - 2 = 12 - 2 = 10$.
So, when the input is 4, the output is 10.
- For $g(-1)$: Replace $x$ with -1.
$g(-1) = 3(-1) - 2 = -3 - 2 = -5$.
So, when the input is -1, the output is -5.

2. Determine the domain and range.
- For $g(x) = 3x - 2$, there's no number you can't multiply by 3 or subtract 2 from. So, the domain is all real numbers, which you can write as $(-\infty, \infty)$ or $\mathbb{R}$.
- Since you can get any real number as an output by choosing the right $x$, the range is also all real numbers, $(-\infty, \infty)$ or $\mathbb{R}$.

3. Sketch its graph.
Let's find a few points:
- $x = 0 \implies g(0) = 3(0) - 2 = -2$. Plot $(0, -2)$.
- $x = 1 \implies g(1) = 3(1) - 2 = 1$. Plot $(1, 1)$.
- $x = 2 \implies g(2) = 3(2) - 2 = 4$. Plot $(2, 4)$.
- $x = -1 \implies g(-1) = -5$. Plot $(-1, -5)$.

If you plot these points on graph paper and connect them, you'll see a straight line going upwards. This confirms it's a linear function. A vertical line would only ever cross this line once, so it passes the Vertical Line Test.

4. Key Takeaways

• A function assigns exactly one output to each input.
• $f(x)$ is notation for a function, where $x$ is the input and $f(x)$ is the output (also represented as $y$).
• The domain is the set of all possible inputs; the range is the set of all possible outputs.
• A graph is a visual representation of a function's input-output pairs.
• The Vertical Line Test helps you quickly identify if a graph represents a function.
• Not all relationships are functions; some inputs might lead to multiple outputs.

Common Mistakes to Avoid:
- Don't confuse $f(x)$ with "f times x"; it means "the function $f$ evaluated at $x$".
- Forgetting that a function must have only one output for any given input.
- Assuming the domain and range are always all real numbers; look for restrictions (like square roots or denominators).
- Misinterpreting graphs: always remember the horizontal axis is typically input and the vertical is output.

5. Now Try It

Think about the function $h(x) = \frac{1}{x-3}$.
1. Calculate $h(0)$, $h(2)$, and $h(4)$.
2. What value of $x$ would you not be able to plug into this function? Why? This will help you define its domain.
3. Based on your understanding, what do you think the range of this function would be (what outputs can you never get)?
4. Does this function pass the Vertical Line Test if you were to graph it?
Success looks like correctly identifying the restriction on the input, calculating the outputs for the given inputs, and making a reasonable guess about the range.