Domain of a Function Game

Domain of a Function Game
This game will require you to find the domain of a function. Scroll down for a detailed explanation.
 

 

How to Play the Domain of a Function Game
There are four types of problems in this game:

1. Polynomials
   The Logic: Polynomials (like x² + 5).
   You can plug any number into them—positive, negative, or zero—and they will always give you a result.
   The Rule: There are no restrictions.
   The Answer: Always (-∞, ∞).
   

2. Basic Rational Functions
   The Logic: These look like fractions, such as \(\frac{1}{x - 3}\).
   In math, the “bottom” (denominator) can never be zero because division by zero is undefined.
   The Goal: Find the value of x that makes the bottom 0 and exclude it.
   Example: For \(\frac{1}{x - 3}\), x cannot be 3.
   The Answer: Look for the “Union” (∪) symbol that skips exactly that one number:
   (-∞, 3) ∪ (3, ∞).
   

3. Factorable Quadratics
   The Logic: You are given a quadratic in the denominator, like \(\frac{1}{x^2 - 5x + 6}\).
   You must factor it to find the two values that break the function.
   The Step: Factor the bottom. x² - 5x + 6 becomes (x - 2)(x - 3).
   The Goal: x cannot be 2 and x cannot be 3.
   The Answer: Look for the choice that skips two separate holes: (-∞, 2) ∪ (2, 3) ∪ (3, ∞).
   

4. Radical Functions
   The Logic: These involve square roots, like \(\sqrt{x + 4}\).
   In the real number system, you cannot take the square root of a negative number.
   The Rule: The expression inside the root must be ≥ 0.
   The Step: Solve the inequality. For \(\sqrt{x + 4}\), you need x + 4 ≥ 0, which means x ≥ -4.
   The Answer: Look for the bracket [ (which means “including”) starting at that value: [-4, ∞).
   

How to find the domain of a function?
Finding the domain of a function is essentially an act of mathematical detective work. You aren’t looking for what x can be; you are looking for what x is not allowed to be.
In the world of real numbers, there are three main situations that restrict a domain. If your function doesn’t have these, the domain is usually all real numbers, written as (-∞, ∞).

1. The Denominator Rule (Fractions)
   The most famous rule in math: You cannot divide by zero. If x is in the denominator, you must find the values that make that denominator zero and leave them out of the domain.
   The Process: Set the denominator equal to zero and solve for x.
   Example: \(f(x) = \frac{5}{x - 3}\)
   Set x - 3 = 0 → x = 3.
   Domain: Everything except 3.
   Notation: (-∞, 3) ∪ (3, ∞)
   

2. The Even Root Rule (Radicals)
   You cannot take the square root (or any even root) of a negative number and get a real result. The expression inside the radical must be zero or positive. The Process: Take the expression inside the root, set it ≥ 0, and solve the inequality.
   Example: \(f(x) = \sqrt{x + 4}\)
   Set x + 4 ≥ 0 → x ≥ -4.
   Domain: All numbers from -4 upwards.
   Notation: [-4, ∞)
   

3. The Logarithm Rule
   Logarithms are even pickier than square roots. You cannot take the log of zero or a negative number. The input must be strictly positive.
   The Process: Set the argument (the stuff inside the log) > 0 and solve.
   Example: f(x) = ln(x - 2)
   Set x - 2 > 0 → x > 2.
   Domain: All numbers strictly greater than 2.
   Notation: (2, ∞)
   

This video gives a clear, step-by-step approach to find the domain of a function.

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