In mathematics, the domain is sometimes referred to as the domain of a function. The domain of a function is the complete set of possible values of the independent variable. How do you write the domain and the range? Domain and range are written in set builder notation. Can a function have an empty domain? No, a function cannot have an empty set as the domain or empty domain. The domain is valid only if it contains some value in it.
More Important Topics. Learn from the best math teachers and top your exams. Live one on one classroom and doubt clearing. Practice worksheets in and after class for conceptual clarity. Book a Free Session. The perimeter of a circular arena. The position of a moving object. Simple interest. Time, principal, and the interest rate. Example 1. You can also talk about the domain of a relation , where one element in the domain may get mapped to more than one element in the range.
Here, the relation is given as a set of ordered pairs. Note that the domain elements 1 and 2 are associated with more than one range elements, so this is not a function.
But, more commonly, and especially when dealing with graphs on the coordinate plane, we are concerned with functions, where each element of the domain is associated with one element of the range. See The Vertical Line Test. The range is also all real numbers except zero.
Domains can also be explicitly specified, if there are values for which the function could be defined, but which we don't want to consider for some reason. We learn about sin and cos graphs later in Graphs of sin x and cos x. Note 1: Because we are assuming that only real numbers are to be used for the x -values, numbers that lead to division by zero or to imaginary numbers which arise from finding the square root of a negative number are not included.
The Complex Numbers chapter explains more about imaginary numbers, but we do not include such numbers in this chapter. Note 2: When doing square root examples, many people ask, "Don't we get 2 answers, one positive and one negative when we find a square root?
See this discussion: Square Root 16 - how many answers? Note 3: We are talking about the domain and range of functions , which have at most one y -value for each x -value, not relations which can have more than one. It's always a lot easier to work out the domain and range when reading it off the graph but we must make sure we zoom in and out of the graph to make sure we see everything we need to see. However, we don't always have access to graphing software, and sketching a graph usually requires knowing about discontinuities and so on first anyway.
In the numerator top of this fraction, we have a square root. In general, we determine the domain by looking for those values of the independent variable usually x which we are allowed to use.
We have to avoid 0 on the bottom of a fraction, or negative values under the square root sign. The range is found by finding the resulting y -values after we have substituted in the possible x -values. There would be a 0 on the bottom of the fraction. Range: No matter how large or small t becomes, f t will never be equal to zero.
There are no resulting square roots of negative numbers or divisions by zero involved here. The function is part of a parabola. In case you missed it earlier, you can see more examples of domain and range in the section Inverse Trigonometric Functions. We fire a ball up in the air and find the height h , in metres, as a function of time t , in seconds, is given by. Generally, negative values of time do not have any meaning. Also, we need to assume the projectile hits the ground and then stops - it does not go underground.
So we need to calculate when it is going to hit the ground. So we solve:. We can see from the function expression that it is a parabola with its vertex facing up. This makes sense if you think about throwing a ball upwards.
It goes up to a certain height and then falls back down. What is the maximum value of h? We use the formula for maximum or minimum of a quadratic function. By observing the function of h , we see that as t increases, h first increases to a maximum of Sometimes we don't have continuous functions. What do we do in this case?
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