Question

Consider the function ()=√+tan . Which of the following statements is true?

A) () is defined for all real values of .
B) () is only defined for positive values of .
C) () is undefined when tan = −4.
D) The range of () is all real numbers.
E) () is an even function.

Answer 1

The given function is not defined for values of x where tan(x) is undefined. The function is undefined when tan(x) = -4. The function is not an even function.

Step-by-step explanation:

The given function is ()=√x+tan(x). Let's analyze the statements to determine which one is true.

A) () is defined for all real values of x: This statement is false because the function is not defined for values of x where tan(x) is undefined, such as x = π/2 + nπ, where n is an integer.

B) () is only defined for positive values of x: This statement is false because the function is defined for both positive and negative values of x.

C) () is undefined when tan(x) = -4: This statement is true. We can solve tan(x) = -4 by taking the inverse tangent function on both sides: x = tan^(-1)(-4). However, the inverse tangent function is not defined when the argument is -4, so the function () is undefined at this point.

D) The range of () is all real numbers: This statement is false because the function is not defined for values of x where tan(x) is undefined.

E) () is an even function: This statement is false because an even function should satisfy () = (), but in this case, () = √x+tan(x) ≠ √(-x)+tan(-x) = -√x+tan(x).