Constraints (Meta) Math Example 3

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Example 3

easy
State the domain constraint for f(x)=x−4f(x) = \sqrt{x - 4} and solve for xx when f(x)=3f(x) = 3.

Solution

  1. 1
    Constraint: the radicand must be non-negative, so x−4≥0x - 4 \ge 0, giving x≥4x \ge 4.
  2. 2
    Set x−4=3\sqrt{x-4} = 3: square both sides to get x−4=9x - 4 = 9, so x=13x = 13.
  3. 3
    Check: 13≥413 \ge 4. Valid.

Answer

x=13,domain constraint: x≥4x = 13,\quad \text{domain constraint: } x \ge 4
Square root functions are only defined for non-negative radicands. The domain constraint must be stated and verified for any solution.

About Constraints (Meta)

Constraints are conditions, rules, or boundaries that restrict which values or solutions are allowed in a mathematical problem, narrowing an infinite space of possibilities to a manageable set.

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More Constraints (Meta) Examples

Example 1 easy

Solve [formula] and identify all constraints on [formula] before solving.

Example 2 medium

An integer [formula] satisfies two constraints: [formula] and [formula], and also [formula] is prime

Example 4 medium

A rectangle has area 36 cm² and integer side lengths. List all constraints and find all valid dimens

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