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In general, we determine the domainof each function by looking for those values of the independent variable (usually x) which we are allowed to use. The curve goes on forever vertically, beyond what is shown on the graph, so the range is all non-negative values of `y`. From the calculator experiment, and from observing the curve, we can see the range is y betweeen −1 and 1.
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 range is found by finding the resulting y-values after we have substituted in the possible x-values. In case you missed it earlier, you can see more examples of domain and range in the section Inverse Trigonometric Functions. Latest Functions and Graphs forum posts:Domain and range [Solved!] by shaikshavali How to obtain 0.84?
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. As you noticed, sometimes it's easier just to be able to visualize things and use intuition rather than working through all of the algebra.

Plug $y = x^2$, the endpoints, into the equation, you get $f(x, y) = \sqrt 0$, so one endpoint of the range is zero. If it were possible for the equation to give negative values, we'd have to check critical points to see what the minimum negative value is, but a square root can't be negative so there is no need. If the equation had been $g(x) = x^2 + 2x - 6$ then for the range we'd get "up to infinity" from checking endpoints and "down to -7" by checking the critical point at (-1, -7). Not the answer you're looking for?Browse other questions tagged multivariable-calculus graphing-functions or ask your own question. Will altering my HOSTS file allow me to access sites that can't be directly accessed via IP? Is it acceptable for a student to ask a professor to be included as a co-author on the professor's paper? Discouraged from giving A's, but students are doing excellent work, what can a professor do? Note: If you intend to use an image you find here for commercial use, please be aware that some photos do require a model or property release. A step by step tutorial on how to determine the properties of the graph of cubic functions and graph them.

If the leading coefficient is positive, as x increases the graph of f is up and as x decreases indefinitely the graph of f is down. If the leading coefficient is negative, as x increases the graph of f is down and as x decreases indefinitely the graph of f is up. Since the leading coefficient (of x3) is positive, the graph of f is up on the right and down on the left and hence the range of f is the set of all real numbers. The Complex Numbers chapter explains more about imaginary numbers, but we do not include such numbers in this chapter. Properties, of these functions, such as domain, range, x and y intercepts, zeros and factorization are used to graph this type of functions. Extreme care was taken to enhance all aspects of her comfort and style, while maintaining her pedigree as a true performance yacht.

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