Question

If an exponential function is of the form f(x) = bx^c, and b > 1, then it is ______ as x approaches infinity. If a logarithmic function is of the form f(x) = log_b(x) + c, and b > 1, then it is _______ as x approaches infinity.

A) Exponential, Increasing.
B) Exponential, Decreasing.
C) Logarithmic, Increasing.
D) Logarithmic, Decreasing.

Answer 1

The exponential function f(x) = bx^c, with b > 1, is increasing as x approaches infinity. The logarithmic function f(x) = log_b(x) + c, with b > 1, is also increasing as x approaches infinity. Hence, the answer is Logarithmic, Increasing.

Step-by-step explanation:

If an exponential function is of the form f(x) = bx^c, and b > 1, then it is increasing as x approaches infinity. For the exponential function, as x increases, the power to which the greater-than-one base is raised also increases, resulting in the entire expression increasing without bound. Thus, the complete function tends toward infinity as x gets larger.

If a logarithmic function is of the form f(x) = log_b(x) + c, and b > 1, then it is increasing as x approaches infinity. This is because logarithms with bases greater than one will increase as the input value x increases, albeit at a decreasing rate. Nevertheless, the output of the function will continue to grow without bound as x approaches infinity.

Therefore, the correct option is C) Logarithmic, Increasing.