LEMMA

Functions

Function domain and range

ALEKS placement

The domain is every input the rule allows — find it by excluding what breaks.

What this covers

  • Set any denominator equal to zero and exclude those inputs.
  • Require anything under an even root to be at least 0.
  • Require any logarithm's argument to be strictly positive.
  • Range often comes from the graph's shape, especially a vertex, an endpoint, or an asymptote.
  • An asymptote is a line the graph approaches as the inputs keep changing. A graph can cross a horizontal or slant asymptote, so inspect the actual range.

Worked example

Worked example

Find the domain of f(x) = √(x - 2)/(x - 5).

  1. x−2≥0⇒x≥2An even root cannot take a negative input.
  2. x=5The denominator cannot be zero.
  3. [2,5)∪(5,∞)Both conditions must hold at once.
Another worked example

Find the range of g(x) = (x - 2)² + 1 over all real x.

  1. (x−2)2≥0A real number squared cannot be negative.
  2. g(x)≥1Adding 1 to a nonnegative square gives an output of at least 1.
  3. g(2)=(2−2)2+1Test input 2 to see whether the rule actually reaches that lower bound.
  4. g(2)=1The square is zero, leaving output one. On the graph, this point has horizontal coordinate two and vertical coordinate one.
  5. range=[1,∞)The square grows without bound, so every output from 1 upward is possible.

A common mistake

A mistake Lemma catches

From x+3>0

x>3x>−3

“To clear +3, subtract 3 from both sides: x > 0 − 3 = −3. Adding 3 instead gives x > 3, which wrongly leaves out inputs such as x = 0, where log(0 + 3) is defined. The boundary is where x + 3 is 0: −3 + 3 = 0, not 3 + 3 = 6.”

Try one

Sample problem

Which x-value is excluded from the domain? Enter an inequality such as x ≠ a.

f(x)=x+51​

Practice domain & range free

No account, no email. Every line you type is checked by the same computer algebra system as the worked example above.

Where this fits

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