الرياضيات
What Is the Domain of a Function?
The domain is every input a function accepts. The two things that exclude a value — a zero denominator and a negative under a root — worked and written out.
Domain of a function
The set of input values a function is allowed to take — every x for which the rule produces a real answer.
- أين يقابله الطلاب
- Grade 9 or 10 mathematics, once functions are written in f(x) notation, and on the Digital SAT in questions asking which values are not permitted.
الإجابة باختصار
The domain of a function is the set of inputs it accepts. Unless a question restricts it, the domain is every real number except the ones that break the rule: values that make a denominator zero, and values that put a negative number under a square root.
مثال
f(x) = 1/(x − 3) → x ≠ 3 g(x) = √(x − 2) → x ≥ 2
For f, the denominator is zero when x = 3, and dividing by zero produces nothing at all, so 3 is the single value the function will not take. Everything else is allowed. For g, the expression under the root must not be negative, so x − 2 ≥ 0 and therefore x ≥ 2. The sign is worth pausing on: it is greater than or equal to, because x = 2 gives √0 = 0, which is a perfectly ordinary output.
The two things that rule a value out
At GCSE and IGCSE level, almost every domain question comes down to one of two problems. Either a value would make a denominator zero, or it would ask for the square root of a negative number. Scan the rule for a fraction and a root sign, and if neither is present the domain is usually every real number.
The method is the same in both cases: write down the condition the expression demands, then solve it as an inequality. A denominator gives x − 3 ≠ 0, so x ≠ 3. A root gives x − 2 ≥ 0, so x ≥ 2. The answer to a domain question is an inequality, not a number.
Watch for both appearing together. In 1/√(x − 2) the root demands x ≥ 2 and the denominator forbids the value that makes the root zero, so the domain is x > 2 and the boundary point is excluded after all.
Domains that are handed to you
Sometimes there is nothing to work out because the question has already decided. f(x) = x² for 0 ≤ x ≤ 5 is a deliberately shortened function, and the restriction is part of its definition rather than a hint about it.
Restricting a domain is not decoration. It changes what the function can output, and it can change what the function is capable of. f(x) = x² over all real numbers cannot be inverted, because 9 would have to come back as both 3 and −3. The same rule restricted to x ≥ 0 can be, and that restriction is why √ has a single agreed value.
Context supplies restrictions too. A function giving the area of a rectangle in terms of one side has a domain limited by the fact that a length cannot be negative, whatever the algebra would tolerate.
How to write the answer down
Inequality form is the safest: x ≥ 2, or x ≠ 3, or 0 ≤ x ≤ 5. Set notation such as {x : x ≥ 2} appears in some textbooks and means the same thing, and writing all real numbers except 3 in words is accepted in most mark schemes.
The mistake to avoid is answering a different question. The domain is what may go in; the roots are the inputs that produce zero; the range is what comes out. A student who answers x = 3 to a domain question about 1/(x − 3) has found the one value that is definitely not in the domain.
أسئلة شائعة
What is the domain of 1/(x − 3)?
Every real number except 3, written x ≠ 3. At x = 3 the denominator becomes zero and the function has no value to return. Every other input, including large negatives and numbers extremely close to 3, works perfectly well — it is a single point removed from the number line, not a region.
Is the domain of √x written x > 0 or x ≥ 0?
x ≥ 0. Zero is allowed, because √0 = 0 is a real answer like any other. Only negatives are excluded, since no real number squares to give a negative. Losing the equals part of the sign is a common single-mark error, and it appears in almost every root-based domain question.
What is the difference between domain and range?
Domain is what goes in, range is what comes out. On a graph the domain is measured along the x-axis and the range along the y-axis. Questions often ask for both in one part, and answering them the wrong way round loses both marks even though the working was right.
Why does the domain matter when finding an inverse?
Because a function can only be inverted if no two inputs share an output. f(x) = x² fails on all real numbers, since 3 and −3 both give 9 and the inverse would not know which to return. Restricting the domain to x ≥ 0 removes the ambiguity and makes the inverse well defined.
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