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This is particularly true with rational and radical functions, which can have restrictions to domain, range, or both.Other functions, such as quadratic functions and polynomial functions of even degree, also can have restrictions to their range.The line of reflection here is x = 1, so the vertex must be at the point (1, f(1)).
For example, if x = −4, then −x = −(−4) = 4, a positive number.
Remember, here the range is restricted to all real numbers.
When defining a function, you usually state what kind of numbers the domain (x) and range (f(x)) values can be.
But even if you say they are real numbers, that doesn’t mean that all real numbers can be used for x.
It also doesn’t mean that all real numbers can be function values, f(x). The restrictions partly depend on the type of function.
could happen whenever the function has a variable in the denominator of a rational expression.
Wolfram|Alpha is a great tool for finding the domain and range of a function.
It also shows plots of the function and illustrates the domain and range on a number line to enhance your mathematical intuition.
Although a function may be given as “real valued,” it may be that the function has restrictions to its domain and range.
There may be some real numbers that can’t be part of the domain or part of the range.