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The term convex function is used in mathematics. It is used to define an interval where the line segment between two points is above the graph, this can both be downward or upward.
Wherever a function is differentiable, it must also be continuous. The opposite is not true, however. For example, the absolute value function, f(x) =|x|, is not differentiable at x=0 even though it is continuous everywhere.
The range of the arcsinx function is restricted because it is the inverse of a function that is not one-to-one, a characteristic usually required for a function to have an inverse. The reason for this exception in the case of the trigonometric functions is that if you take only a piece of the function, one that repeats through the period and is able to represent the function, then an inverse is obtainable. Only a section that is one-to-one is taken and then inverted. Because of this restriction, the range of the function is limited.
The domain of f(x)=3sin(2x) is all real numbers ----Any number can be input into this function and receive a valid output The range of f(x)=3sin(2x) is [-3,3] ----The range of y=sin(x) is [-1,1] frequency modulation, which happens when the argument of a sine function is modified, does not affect the range of a cosine or sine function, so the range of y=sin(2x) is also [-1,1]. Amplitude modulation, which happens when the entire function is multiplied by a numerical constant, does affect the range. If any number put into y=sin(2x) will output a maximum of 1, the most an input can cause in y=3sin(2x) will be 3 times the maximum of y=sin(2x), and the same for the minimums, so the range of y=3sin(2x) is from -3 to 3. If you would like a more complete explanation of the concepts underlying domain and range of functions, message me and I can more completely explain them.
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