Question

Which statement is true about the functions f(x) = square root of (2x) and g(x) = (x^2 - 2)/2?

A) Both functions are quadratic.
B) f(x) is the square root of g(x).
C) g(x) is the derivative of f(x).
D) f(x) + g(x) equals x^2 - 1.

Answer 1

None of the statements about the functions f(x) = square root of (2x) and g(x) = (x^2 - 2)/2 are correct. Statement D appeared to be correct, but after verifying by adding f(x) and g(x) together, it was determined to be false as well.

Step-by-step explanation:

Considering the given functions f(x) = √(2x) and g(x) = (x^2 - 2)/2, we can evaluate the statements provided:

• A) Incorrect. Only g(x) is quadratic. f(x) is a square root function, not a quadratic.
• B) Incorrect. f(x) is not the square root of g(x), they are independent functions.
• C) Incorrect. g(x) is not the derivative of f(x). The derivative of f(x) would involve a fraction with variables in the denominator.

• D) Correct. When you add f(x) and g(x) together, you get f(x) + g(x) = x^2 - 1.

To verify option D, let's add both functions:

f(x) + g(x) = √(2x) + rac{x^2 - 2}{2} = √(2) * √(x) + rac{1}{2}x^2 - 1

Since the product of a number and its square root, such as √(2) * √(x), results in that number to the power of 1, i.e., 2·x. However, notice there's an error in simplification above; hence let's get the correct simplification:

f(x) + g(x) = √(2x) + rac{x^2 - 2}{2} = x + rac{x^2}{2} - 1 = rac{2x + x^2 - 2}{2}

This does not simplify to x^2 - 1, indicating that an error occurred in this verification, and option D is also incorrect.

Therefore, all given statements about the functions are false.