الرياضيات
What Is the Range of a Function?
The range is what a function can actually output. Found from the graphs of y = x² + 1 and y = 1/x, and told apart from the domain on a mapping diagram.
Range of a function
The set of output values a function actually produces once every input in its domain has been used.
- أين يقابله الطلاب
- Grade 9 or 10 mathematics, always in the same lesson as domain, and on IGCSE Extended papers where both are asked for in one question.
الإجابة باختصار
The range of a function is the set of values it actually outputs. For f(x) = x² + 1 the range is f(x) ≥ 1, because a square is never negative and the smallest input contribution is zero. For f(x) = 1/x it is every real number except 0, since no input produces zero.
مثال
f(x) = x² + 1 → f(x) ≥ 1 f(x) = 1/x → f(x) ≠ 0
Squaring any real number gives zero or more, so x² + 1 is at least 1, and it hits exactly 1 when x = 0. That gives the range f(x) ≥ 1, with the equals sign included because the value is genuinely reached. For 1/x the argument runs differently: dividing 1 by anything, however large, never produces exactly zero, though the outputs get arbitrarily close. Every other real number appears, so the range is everything except 0.
Read it off the vertical axis
Sketch the graph and the range is the set of heights the curve occupies. Domain is a question about how far the picture spreads sideways; range is a question about how far it spreads up and down. Asking which y values are used is usually faster than any algebra.
y = x² + 1 sits on top of its lowest point, (0, 1), and climbs from there in both directions, so every height from 1 upwards is used and nothing below. y = 1/x is in two separate pieces, one above the axis and one below, and between them they use every height except the one the curve never reaches.
For a curve without a lowest point the answer is simply all real numbers. Any non-horizontal straight line, such as y = 3x − 4, has that range: continue far enough in either direction and every height is eventually reached.
What a mapping diagram shows, and what it does not
On a mapping diagram the inputs are drawn on the left and the outputs on the right, with arrows between them. The left-hand column is the domain. The range is not the right-hand column — it is only the part of it that has an arrow pointing at it.
That distinction has a name. The set you draw on the right is the codomain, the values the function is permitted to produce; the range is the values it actually produces. For f(x) = x² you might write all real numbers on the right, but no arrow ever lands on a negative, so the range is f(x) ≥ 0. Exam questions want the values reached.
Change the domain and the range moves
The two are tied together, which is why questions hand you one and ask for the other. f(x) = x² over every real number has range f(x) ≥ 0. The same rule restricted to 1 ≤ x ≤ 3 has range 1 ≤ f(x) ≤ 9, because the outputs now stop at both ends.
For a restricted domain, the safe method is to evaluate the function at both endpoints and then check whether a turning point sits between them. For f(x) = x² − 6x + 5 on 0 ≤ x ≤ 5 the endpoints give 5 and 0, but the minimum at x = 3 gives −4, so the range runs from −4 up to 5. Skipping the turning point is how the wrong answer gets written.
أسئلة شائعة
How do you find the range without drawing the graph?
Ask what the rule can and cannot produce. A squared term is never negative, so anything of the form x² + k has range at least k. A fraction with 1 on top never gives zero. Completing the square finds the lowest or highest value of a quadratic exactly, which is usually all the range question needs.
Is the range the same as the codomain?
No. The codomain is the set of values a function is allowed to output; the range is the set it actually outputs. For f(x) = x² the codomain might be all real numbers, but the range is only f(x) ≥ 0. School questions almost always want the range, so the values actually reached.
What is the range of a linear function like y = 3x − 4?
All real numbers. A straight line with a non-zero gradient keeps rising and falling without limit, so every height is used somewhere along it. The exception is a horizontal line such as y = 2, whose range is the single value 2, since that is the only output it ever gives.
Should the answer use y or f(x)?
Either is accepted, but stay consistent with how the question was written. If it defines f(x) = x² + 1, answer f(x) ≥ 1. If it gives y = x² + 1, answer y ≥ 1. Writing x ≥ 1 answers a domain question by mistake and is the most common way the mark is lost.
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