Restricted Domain Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Restricted Domain.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Restricting a domain limits allowable inputs so a function has desired properties, often invertibility.

You keep only the input interval where the function behaves one way.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: A good inverse may require narrowing the original input set.

Common stuck point: Students restrict outputs instead of inputs when creating inverses.

Sense of Study hint: Check monotonic intervals and choose one that matches the target range.

Worked Examples

Example 1

easy
Find the natural (implied) domain of f(x) = \sqrt{x - 3}.

Solution

  1. 1
    The square root function requires a non-negative argument: x - 3 \ge 0.
  2. 2
    Solve: x \ge 3.
  3. 3
    The domain is [3, \infty).

Answer

[3, \infty)
A restricted domain limits the inputs of a function to a subset of all real numbers. For square roots, the expression under the radical must be non-negative. For fractions, the denominator cannot be zero. These natural restrictions come from the mathematical definition of the operations involved.

Example 2

medium
Restrict the domain of f(x) = x^2 so that the function has an inverse. Find the inverse on this restricted domain.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium
Find the domain of g(x) = \frac{\sqrt{x+4}}{x^2 - 9}.

Example 2

hard
Restrict the domain of f(x) = (x - 1)^2 + 3 to the largest interval containing x = 4 on which f is one-to-one, then find f^{-1}(x).
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Background Knowledge

These ideas may be useful before you work through the harder examples.

domainfunction definitioninverse function