Question

attempt 1: 10 attempts remaining. which of the following functions is the only one that is continuous over the entire real line? a. ( e^{x^2} ) b. ( sqrt{2x} ) c. none of these functions d. ( ln(x + 1) ) submit answer next item

Explanation:

• For option A: The function \( y = e^{x^2} \) is a composition of the exponential function \( e^u \) (which is continuous everywhere) and \( u = x^2 \) (which is a polynomial, hence continuous everywhere). By the composition of continuous functions theorem, \( e^{x^2} \) is continuous for all real numbers \( x \).
• For option B: The function \( y=\sqrt{2x} \) has a domain where \( 2x\geq0 \), i.e., \( x\geq0 \). So it is not defined (and thus not continuous) for \( x < 0 \).
• For option D: The function \( y = \ln(x + 1) \) has a domain where \( x+ 1>0 \), i.e., \( x>- 1 \). It is not defined (and thus not continuous) for \( x\leq - 1 \).

Answer:

A. \( e^{x^2} \)