Question

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) )

Explanation:

Step1: Analyze Option A ($e^{x^2}$)

The exponential function $y = e^u$ is continuous for all real $u$, and $u = x^2$ is a polynomial (hence continuous for all real $x$). The composition of two continuous functions is continuous, so $e^{x^2}$ is continuous for all real $x$.

Step2: Analyze Option B ($\sqrt{2x}$)

The square - root function $\sqrt{t}$ is defined when $t\geq0$. For $\sqrt{2x}$, we need $2x\geq0\Rightarrow x\geq0$. So, this function is not defined (and thus not continuous) for $x < 0$.

Step3: Analyze Option D ($\ln(x + 1)$)

The natural logarithm function $\ln(t)$ is defined when $t>0$. For $\ln(x + 1)$, we need $x + 1>0\Rightarrow x>- 1$. So, this function is not defined (and thus not continuous) at $x\leq - 1$.

Answer:

A. $e^{x^2}$