Functions: Domain and Range Fundamentals

TL;DR

The domain of a function is all the possible input values (x-values) it can take, while the range is all the possible output values (y-values) it can produce. You need to identify restrictions on inputs, like division by zero or taking the square root of a negative number, to find the domain. The range then depends on these valid inputs and the function's behavior.

1. The Mental Model

Think of a function as a machine: you put something in (the domain), the machine processes it, and something comes out (the range). The machine might have rules about what it can accept as input, and those rules define its domain.

2. The Core Material

When we talk about functions, the domain is the set of all possible input values (often 'x') for which the function is defined. The range is the set of all possible output values (often 'y' or 'f(x)') that the function can produce.

Identifying Domain Restrictions

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For most functions you'll encounter, especially at this level, you're looking for specific issues that would make the function undefined:

1. Division by zero: You can't divide by zero. So, if your function has a fraction, any input that makes the denominator zero is not in the domain.
2. Even roots of negative numbers: You can't take the square root (or fourth root, sixth root, etc.) of a negative number and get a real number result. So, any input that makes the expression inside an even root negative is not in the domain.
3. Logarithms of non-positive numbers: You can't take the logarithm of zero or a negative number. So, any input that makes the argument of a logarithm zero or negative is not in the domain.

If none of these restrictions apply, the domain is usually all real numbers (ℝ).

Determining the Range

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Finding the range can be a bit trickier than finding the domain. Once you have the domain, you can often visualize the graph of the function or use your knowledge of parent functions to see what y-values are possible.

• For linear functions (e.g., y = 2x + 1): The range is typically all real numbers, unless the domain is restricted.
• For quadratic functions (e.g., y = x²): The range will have a minimum or maximum value at the vertex. For y = x², the lowest y-value is 0, so the range is [0, ∞).
• For square root functions (e.g., y = √x): The output is always non-negative. If the domain is [0, ∞), the range is also [0, ∞).
• For rational functions: As x approaches values not in the domain (vertical asymptotes), y often approaches infinity or negative infinity. As x approaches infinity, y might approach a horizontal asymptote.

Here's how to think about the process:

graph TD
    A["Start with the function"] --> B{"Any fractions?"};
    B -- Yes --> C{"Denominator = 0?"};
    C -- Yes --> D["Exclude x-values that make denominator zero"];
    B -- No --> E{"Any even roots (√, ⁴√, etc.)?"};
    E -- Yes --> F{"Expression inside root < 0?"};
    F -- Yes --> G["Exclude x-values that make expression negative"];
    E -- No --> H{"Any logarithms (log, ln)?"};
    H -- Yes --> I{"Argument ≤ 0?"};
    I -- Yes --> J["Exclude x-values that make argument zero or negative"];
    H -- No --> K["No obvious restrictions"];
    D --> L["Domain is all other real numbers"];
    G --> L;
    J --> L;
    K --> L;
    L --> M["Find Domain"];
    M --> N["Consider graph behavior or transformations"];
    N --> O["Identify all possible output (y) values"];
    O --> P["Determine Range"];

Writing Domain and Range

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You'll usually write domain and range using interval notation or set-builder notation.

• Interval Notation: Uses parentheses () for values not included and square brackets [] for values that are included.
  • (-∞, ∞) means all real numbers.
  • [0, ∞) means all real numbers greater than or equal to 0.
  • (-3, 5] means all numbers between -3 and 5, including 5 but not -3.
• Set-Builder Notation: Describes the set using a rule.
  • {x | x ∈ ℝ, x ≠ 0} means "all real numbers x such that x is not equal to 0."

3. Worked Example

Problem: Find the domain and range of the function $f(x) = \sqrt{x - 4}$.

Domain:
1. This function has an even root (square root).
2. The expression inside the square root, $(x - 4)$, must be greater than or equal to zero.
3. So, we set up the inequality: $x - 4 \ge 0$.
4. Adding 4 to both sides gives: $x \ge 4$.
5. Therefore, the domain is all real numbers greater than or equal to 4.
* In interval notation: $[4, \infty)$
* In set-builder notation: $\{x | x \in ℝ, x \ge 4\}$

Range:
1. We know the smallest value $x-4$ can be is 0 (when $x=4$).
2. The square root of 0 is 0. So, the smallest possible output of $f(x)$ is 0.
3. As $x$ increases from 4, $x-4$ increases, and $\sqrt{x-4}$ also increases.
4. There's no upper limit to how large $x$ can be, so there's no upper limit to how large $\sqrt{x-4}$ can be.
5. Therefore, the range is all non-negative real numbers.
* In interval notation: $[0, \infty)$
* In set-builder notation: $\{y | y \in ℝ, y \ge 0\}$

4. Key Takeaways

• The domain is the set of all allowed input values (x).
• The range is the set of all possible output values (y).
• Look for division by zero (denominator cannot be zero), even roots of negative numbers (expression inside root must be $\ge 0$), and logarithms of non-positive numbers (argument must be $> 0$) to find domain restrictions.
• If no restrictions are found, the domain is usually all real numbers.
• The range often requires visualizing the graph or considering the function's behavior based on its domain.
• Use interval notation or set-builder notation to express domain and range.

Common Mistakes to Avoid:

• Forgetting to consider all restrictions (e.g., only checking for square roots but missing a denominator).
• Confusing domain (x-values) with range (y-values).
• Incorrectly using parentheses vs. square brackets in interval notation (e.g., using (4, ∞) instead of [4, ∞) when 4 is included).
• Assuming the range is always all real numbers; it's often restricted.

5. Now Try It

Exercise: Find the domain and range of the function $g(x) = \frac{1}{x+3}$.

What to do:
1. Identify any potential issues for the domain (like division by zero).
2. Set up an inequality or equation to exclude invalid x-values.
3. Write the domain in interval notation.
4. Consider what y-values are possible and what y-values are impossible (e.g., can the fraction ever equal zero?).
5. Write the range in interval notation.

What success looks like:
Your domain should exclude the value that makes the denominator zero. Your range should exclude zero, as a fraction can only be zero if its numerator is zero, and here the numerator is 1.