The function is continuous for all real numbers

MATHEMATICS · Single correct · AP · Calculus AB · Limits and Continuity · Limits and Continuity

Problem

The function

h(x)=\ln\left(3x^{2}+2\right)

is continuous for all real numbers. Which statement best justifies this conclusion? (a) Because

\ln x

is continuous for all real numbers, replacing

x

with

3x^{2}+2

does not affect continuity. (b) Because

3x^{2}+2\ge 2

for all real numbers,

h(x)

is bounded, and every bounded logarithmic function is continuous for all real numbers. (c) Because

3x^{2}+2>0

for all real numbers, and both the quadratic and the natural logarithm are continuous on their respective domains, the composition

\ln(3x^{2}+2)

is continuous everywhere. (d) Because the derivative of

3x^{2}+2

exists for all

x

, the logarithm of the expression must also be differentiable and therefore continuous.