The range of a function refers to the set of all possible output values. In this case, the range of f(x) is all real numbers greater than or equal to 0, since the exponential function always produces a positive output. The range of g(x) is the same, as the added term of 3ex2 does not affect the overall shape of the graph or the range.
To find the maximum value of a function, we can use the first derivative test. First, we take the derivative of the function and set it equal to 0. Then, we solve for x to find the critical points. Next, we plug these values into the second derivative to determine if they correspond to a maximum or minimum. Finally, we can use these critical points to find the corresponding y-values, which will be the maximum values in this case.
For both of these functions, there are no restrictions on the input values. This means that they can take on any real number as an input and produce a corresponding output. However, it is important to note that as the input values become very large, the outputs will also become very large, so the functions may not be well-defined for extremely large inputs.
Since both functions involve the same exponential term, they have the same range. This means that the ranges of f(x) and g(x) are equal, and both functions have a range of all real numbers greater than or equal to 0.
No, neither of these functions can produce a negative output. The exponential function always produces a positive output, and the added term in g(x) does not change this. Therefore, the range of both functions is limited to only positive values.