Final answer:
Finding the compositions f(g(x)) and g(f(x)) for the functions f(x) = √(x²) and g(x) = x² both result in x², with the caveat that √(x²) simplifies to |x|. The correct option is b) √(x⁴), which simplifies to x², representing the compositions of the functions.
Step-by-step explanation:
The student is asking to find the compositions of two functions f(x) = √(x²) and g(x) = x². Let's figure out the composition f(g(x)) first. By substituting g(x) into f, we get f(g(x)) = f(x²) = √((x²)²) = √(x⁴). Now, √(x⁴) simplifies to x² because the square root of x to the fourth is x squared.
Next, we calculate the composition g(f(x)). By substituting f(x) into g, we get g(f(x)) = g(√(x²)) = (√(x²))². Since √(x²) is equal to the absolute value of x, denoted as |x|, we find that g(f(x)) simplifies to |x|², which is x² again, because x, when squared, is positive regardless of the sign of x.
Therefore, the compositions f(g(x)) and g(f(x)) both simplify to x². Hence, the correct option is b) √(x⁴), which simplifies to x². However, it is key to remember that the principal value of the square root of a square is the absolute value of the original expression, so a better expression for f(x) is |x|. Yet, since the question asked for compositions and not for a simpler form of the function alone, the best answer with the context provided is still x².