Question

For the function f(x) = Vez* + 2e* - 3 - her, what is the domain?

a) All real numbers
b) Only positive real numbers
c) Only negative real numbers
d) Complex numbers

Answer 1

The domain of the function f(x) = √e^{x^2} + 2e^x - 3 - √x is 'Only positive real numbers' because while exponents are defined for all real numbers, the square root of x requires x to be non-negative.

Step-by-step explanation:

The provided function is f(x) = √e^{x^2} + 2e^x - 3 - √x. First, let's identify any restrictions that may apply to the domain of f(x). The domain of a function is the set of all possible input values (x) that the function can accept without resulting in any undefined or complex values.

There are two operations in the function that restrict the domain: the square root and the natural exponentiation (e^x). The natural exponentiation function is defined for all real numbers, so it does not restrict the domain. However, the square root function requires its argument to be non-negative to ensure a real number output. Therefore, x2 is always non-negative, so √e^{x^2} is defined for all real numbers, and e^x is defined for all real numbers. However, √x requires x to be non-negative. As a result, the domain consists only of non-negative real numbers to satisfy the square root operation of x, which means the domain is 'Only positive real numbers'.