Final answer:
The function f(x) = √(x^2 + 3) - x has no vertical asymptotes and has horizontal asymptotes at y = √(x^2 + 3) and y = -√(x^2 + 3).
Step-by-step explanation:
The function f(x) = √(x^2 + 3) - x is a square root function with a linear term subtracted. To find the vertical asymptote(s), we need to determine the values of x for which the function is undefined. The square root function is undefined for negative values, so we set the expression inside the square root equal to 0 and solve for x. x^2 + 3 = 0 has no real solutions, so there are no vertical asymptotes.
To find the horizontal asymptote(s), we examine the behavior of the function as x approaches positive or negative infinity. As x gets very large or very small, the linear term becomes negligible compared to the square root term. Therefore, the function approaches the value of √(x^2 + 3) as x approaches infinity and -√(x^2 + 3) as x approaches negative infinity. Thus, the horizontal asymptote(s) are y = √(x^2 + 3) and y = -√(x^2 + 3).