variable restrictions

• If it's just x\sqrt{x}x​, then x≥0x \geq 0x≥0, because the square root of a negative number isn’t real.

• If it’s a more complicated expression like x2−9\sqrt{x^2 - 9}x2−9​:

  • You solve x2−9≥0x^2 - 9 \geq 0x2−9≥0 to find when it’s non-negative.

  • In this case, x2≥9x^2 \geq 9x2≥9, so x≥3x \geq 3x≥3 or x≤−3x \leq -3x≤−3. This means xxx can be any real number that satisfies those conditions.

• Use this if xxx is under a square root or logarithm (since you can't have negative numbers under these).

• Example: x\sqrt{x}x​, so x≥0x \geq 0x≥0.

• When it's a real number:

  • If there's no square root, fraction, or log, xxx can be any real number.

  • Example: 2x+32x + 32x+3 can have xxx be any real number.

• Multiple variables:

  • Apply the same rule for each variable.

  • Example: x−y\sqrt{x - y}x−y​, means x−y≥0x - y \geq 0x−y≥0, so x≥yx \geq yx≥y.

For 2x+4\sqrt{2x + 4}2x+4​:

• Domain: Solve 2x+4≥02x + 4 \geq 02x+4≥0 to get x≥−2x \geq -2x≥−2, so the domain is [−2,∞)[-2, \infty)[−2,∞).

• Range: The minimum value is at x=−2x = -2x=−2 giving 0=0\sqrt{0} = 00​=0, so the range is [0,∞)[0, \infty)[0,∞).