Domain & Range Calculator
Algebra, Functions
Intro: Analyzes f(x) to determine all allowed x-values (domain) and possible y-values (range), with explanations of exclusions.
Worked example
- Find the domain and range of $f(x) = \sqrt{x-3}$.
- We have $f(x)=\sqrt{x-3}$. The expression inside the square root must be nonnegative.
- Set the radicand condition: $x-3 \ge 0$.
- Solve the inequality: $x \ge 3$.
- Thus the domain is all real x such that $x \ge 3$, i.e., $[3, \infty)$.
- Next, analyze the range. For $x \ge 3$, the radicand $x-3$ is at least 0.
- The square root output is always nonnegative: $\sqrt{x-3} \ge 0$ for any x in the domain.
- As x increases beyond 3, $x-3$ can become arbitrarily large, and so can $\sqrt{x-3}$.
- At the left endpoint x = 3, we have $f(3)=\sqrt{3-3}=0$.
- Therefore, y-values start at 0 and increase without bound: the range is $[0, \infty)$.
- Answer: $\boxed{\text{Domain}=[3,\infty),\; \text{Range}=[0,\infty).}$
FAQs
Can it always find an exact range?
For many standard functions (polynomials, rationals, roots, logs, etc.), yes. Some complicated expressions may need interval checking or graph-based hints.
Why choose MathGPT?
- Get clear, step-by-step solutions that explain the “why,” not just the answer.
- See the rules used at each step (power, product, quotient, chain, and more).
- Optional animated walk-throughs to make tricky ideas click faster.
- Clean LaTeX rendering for notes, homework, and study guides.
How this calculator works
- Type or paste your function (LaTeX like
\sin,\lnworks too). - Press Generate a practice question button to generate the derivative and the full reasoning.
- Review each step to understand which rule was applied and why.
- Practice with similar problems to lock in the method.