Math Buddyv1.0

Domain and range calculator with steps

Type or scan a function. MathBuddy finds its domain and range and writes them in interval notation.

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What this page covers

  • Domains of functions built from square roots, fractions and logarithms, alone or combined.
  • Ranges of quadratics, square roots and other functions whose outputs follow from their shape.
  • Answers in interval notation, with the condition that produced each endpoint.
  • Not covered here: finding the inverse function (use Inverse function calculator) and drawing graphs, which MathBuddy does not produce.

How to enter the problem

  • Type the function, such as f(x) = sqrt(6 − 2x), and say whether you want the domain, the range or both.
  • Scan it. Fit the one problem in the frame.
  • Choose a photo. Several problems in one photo are solved as a list.
  • Then tap Solve.

Worked examples

Example 1

Problem

Find the domain of f(x)=9−x2\displaystyle f(x) = \sqrt{9 - x^2}.

Answer

Verified

The answer is

[−3,3][-3, 3]

Explanation

  1. Write the condition
    A square root needs a non-negative input: 9−x2≥09 - x^2 \ge 0.
  2. Rearrange
    x2≤9x^2 \le 9.
  3. Solve
    −3≤x≤3-3 \le x \le 3, both endpoints included because the root of 0 is 0.

Example 2

Problem

Find the range of g(x)=2(x−1)2+3\displaystyle g(x) = 2(x - 1)^2 + 3.

Answer

Verified

The answer is

[3,∞)[3, \infty)

Explanation

  1. Read the shape
    A parabola opening upward (the 2 is positive) with vertex (1,3)(1, 3).
  2. Solve for xx in terms of yy
    y=2(x−1)2+3y = 2(x - 1)^2 + 3 gives (x−1)2=y−32(x - 1)^2 = \frac{y - 3}{2}.
  3. Find which outputs are possible
    A square is never negative, so y−32≥0\frac{y - 3}{2} \ge 0, which means y≥3y \ge 3.

Example 3

Problem

Find the domain of h(x)=ln⁡(x−1)5−x\displaystyle h(x) = \dfrac{\ln(x - 1)}{\sqrt{5 - x}}.

Answer

Verified

The answer is

(1,5)(1, 5)

Explanation

  1. Logarithm condition
    ln⁡(x−1)\ln(x - 1) needs x−1>0x - 1 > 0, so x>1x > 1.
  2. Root in the denominator
    5−x\sqrt{5 - x} needs 5−x≥05 - x \ge 0, and a denominator cannot be 0, so 5−x>05 - x > 0, which means x<5x < 5.
  3. Combine
    Both conditions must hold at once: 1<x<51 < x < 5.

Common mistakes

  • Allowing zero under a root that sits in a denominator. 5−x\sqrt{5 - x} on the bottom needs 5−x>05 - x > 0, not 5−x≥05 - x \ge 0.
  • Closing a bracket at infinity. Write [3,∞)[3, \infty), never [3,∞][3, \infty].
  • Answering with the domain when the range was asked. For g(x)=2(x−1)2+3g(x) = 2(x - 1)^2 + 3 every real xx is allowed, but the outputs start at 3.
  • Including the boundary of a logarithm. ln⁡(x−1)\ln(x - 1) needs x−1x - 1 strictly positive, so x=1x = 1 is excluded.

Checks, assumptions and limits

  • Test a point on each side of an endpoint. For 6−2x\sqrt{6 - 2x}, x=3x = 3 gives 0=0\sqrt{0} = 0, while x=4x = 4 asks for the root of −2-2, which is not real.
  • Domain and range here are over the real numbers.
  • A round bracket leaves the endpoint out; a square bracket includes it.
  • MathBuddy can make mistakes. Double check important steps.

Related tasks

Frequently asked questions

Yes, by default. Ask a follow-up if you also want it as an inequality or in set-builder notation.

Yes. Ask for the range or for both. The steps show how the range follows from the shape of the function or from solving for x.

Every value that makes a denominator zero is removed from the domain, and the steps list each one.

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