An expression is defined when it gives a real number. Two things break that: dividing by zero and taking the square root of a negative number. Restricting the input means listing the values you must exclude.
This lesson belongs to functions and mappings. It follows reversing a one-to-one mapping, where you saw that a rule such as x² cannot be reversed cleanly over all inputs.
What are the two restrictions?
- A denominator cannot be 0. For 1/(x − 3), the bottom equals 0 when x = 3, so x ≠ 3.
- The value under a square root cannot be negative. For √(x + 5), the inside must satisfy x + 5 ≥ 0, so x ≥ −5.
When both appear in the same rule, both conditions must hold together.
How to find the restrictions, step by step
- Look for fractions. Set each denominator equal to 0 and solve. Those values are excluded.
- Look for square roots. Set the inside as ≥ 0 and solve the inequality.
- Combine all the conditions into one statement.
- Test one excluded value and one allowed value to confirm.
Worked example
State the values of x for which f(x) = √(x − 2)/(x − 5) is defined.
Square root: x − 2 ≥ 0, so x ≥ 2.
Denominator: x − 5 ≠ 0, so x ≠ 5.
Combine: x ≥ 2 and x ≠ 5. That means every number from 2 upwards, except 5.
Test: x = 1 fails the square root because √(−1) has no real value. x = 5 gives 0 on the bottom. x = 6 works: √4/1 = 2.
So f is defined for x ≥ 2, x ≠ 5.
The mistake to watch for
A common slip is to list only one condition when two apply.
Mistaken answer: x ≥ 2
The student dealt with the square root and forgot the denominator.
With x ≥ 2 the value x = 5 is still inside the stated range, and f(5) would need division by 0. The fix is to scan the expression for every fraction line and every root sign before writing the answer. Another slip is reversing the inequality: √(3 − x) needs 3 − x ≥ 0, which gives x ≤ 3, not x ≥ 3.
Check yourself
Try each on paper, then open the answer.
1. For which value of x is 1/(x + 4) not defined?
Show answer
x + 4 = 0 gives x = −4. So the expression is defined for all x except x = −4.
2. State the values of x for which √(3 − x) is defined.
Show answer
3 − x ≥ 0, so 3 ≥ x. The answer is x ≤ 3. Test: x = 2 gives √1 = 1, and x = 4 gives √(−1), which is not real.
3. Which values of x must be excluded from 5/(x² − 9)?
Show answer
x² − 9 = 0 gives x² = 9, so x = 3 or x = −3. The excluded values are x = 3 and x = −3. Check: 3² − 9 = 0 and (−3)² − 9 = 0.
Where this leads next
Restrictions appear again when you meet real situations where only some inputs make sense, in interpreting a function machine in context. The non-calculator working trainer and the percentage-base explorer help you practise spotting when an input or a base cannot be zero.
If you usually get the algebra right but miss the restriction, that habit is easy to train with a teacher in online one-to-one Mathematics tuition.