Functions and Their Graphs

TL;DR

Functions relate inputs to outputs with a unique output for every input, and their graphs are visual representations of this relationship. Understanding domain and range is crucial for defining what goes into and comes out of a function. Different types of functions have distinct graph shapes and properties.

1. The Mental Model

Think of a function as a machine: you put something in (the input), and it gives you exactly one thing back (the output). The graph is like a picture of what that machine does for all possible inputs.

2. The Core Material

A function is a special type of relation where each input (from the domain) corresponds to exactly one output (in the range). This "exactly one output" rule is key.

You can often represent a function using an equation, like $f(x) = x^2$, where $x$ is the input and $f(x)$ is the output.

2.1 Domain and Range

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• Domain: These are all the possible input values ($x$) for which the function is defined. For many functions, the domain is all real numbers ($\mathbb{R}$), but you need to watch out for:
  • Division by zero (e.g., $f(x) = 1/x$, $x \neq 0$)
  • Square roots of negative numbers (e.g., $f(x) = \sqrt{x}$, $x \ge 0$)
• Range: These are all the possible output values ($y$ or $f(x)$) that the function can produce. The range often depends on the function's equation and its domain.

2.2 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 any vertical line intersects the graph at more than one point, it's NOT a function. This is because multiple $y$-values (outputs) for a single $x$-value (input) would violate the function definition.

graph TD
    A["Draw a vertical line"] --> B{"Does it cross the graph more than once?"}
    B -- "Yes" --> C["Not a Function"]
    B -- "No" --> D["It is a Function"]

2.3 Common Types of Functions and Their Graphs

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• Linear Functions: $f(x) = mx + b$. Their graphs are straight lines.
  • Domain: All real numbers.
  • Range: All real numbers (unless $m=0$, then it's a horizontal line, and the range is just $y=b$).
• Quadratic Functions: $f(x) = ax^2 + bx + c$. Their graphs are parabolas (U-shaped curves).
  • Domain: All real numbers.
  • Range: Depends on the vertex of the parabola.
• Polynomial Functions: $f(x) = a_n x^n + ... + a_1 x + a_0$. Smooth, continuous curves.
  • Domain: All real numbers.
• Rational Functions: $f(x) = P(x) / Q(x)$, where $P(x)$ and $Q(x)$ are polynomials. These often have asymptotes (lines the graph approaches but never touches).
  • Domain: All real numbers except where $Q(x) = 0$.
• Radical Functions: $f(x) = \sqrt[n]{g(x)}$.
  • Domain: If $n$ is even, $g(x) \ge 0$. If $n$ is odd, all real numbers.

2.4 Graphing Basics

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To graph a function:
1. Determine the domain.
2. Choose several $x$-values from the domain (including some positive, negative, and zero if appropriate).
3. Calculate the corresponding $y$-values using the function's equation.
4. Plot the $(x, y)$ points.
5. Connect the points with a smooth curve or line, respecting the function's type and domain/range.

3. Worked Example

Let's consider the function $f(x) = \sqrt{x-2}$.

1. Find the domain: We can't take the square root of a negative number. So, the expression inside the square root must be non-negative:
   $x - 2 \ge 0$
   $x \ge 2$
   The domain is $[2, \infty)$.

2. Find the range: Since $\sqrt{\text{anything non-negative}}$ always results in a non-negative number, the smallest output will be $\sqrt{0} = 0$. As $x$ increases, $f(x)$ also increases.
   The range is $[0, \infty)$.

3. Create a table of values and plot:

   • If $x=2$, $f(2) = \sqrt{2-2} = \sqrt{0} = 0$. Point: $(2,0)$
   • If $x=3$, $f(3) = \sqrt{3-2} = \sqrt{1} = 1$. Point: $(3,1)$
   • If $x=6$, $f(6) = \sqrt{6-2} = \sqrt{4} = 2$. Point: $(6,2)$
   • If $x=11$, $f(11) = \sqrt{11-2} = \sqrt{9} = 3$. Point: $(11,3)$

   Plot these points on a coordinate plane and connect them with a smooth curve starting at $(2,0)$ and extending upwards and to the right. Notice how the graph only exists for $x \ge 2$, consistent with our domain.

4. Key Takeaways

• Every input to a function must have exactly one output.
• The domain is all possible input values, while the range is all possible output values.
• The Vertical Line Test helps you visually determine if a graph represents a function.
• Different types of functions (linear, quadratic, etc.) have predictable graph shapes.
• Pay close attention to restrictions on the domain, especially with division or even roots.

• Graphs are powerful visual tools for understanding function behavior.

• Common mistakes:

  • Assuming the domain is always all real numbers.
  • Confusing domain with range.
  • Drawing a graph that fails the Vertical Line Test and calling it a function.
  • Not considering restrictions for square roots (non-negative) or denominators (non-zero).

5. Now Try It

Given the function $g(x) = \frac{1}{x+3}$:
1. Determine its domain and range.
2. Create a small table of values for $x = -5, -4, -2, -1, 0$.
3. Sketch the graph using these points and your understanding of rational functions (think about what happens near $x=-3$).
4. Apply the Vertical Line Test to your sketch.

Success looks like: Correctly identified domain and range, a table with calculated points, and a sketch that clearly shows the function's behavior, including a vertical asymptote at $x=-3$, and passes the Vertical Line Test.