Question

The exponential function f with base b is defined by f(x) = ___, b >0, and b≠1. using interval notation, the domain of this function is ____ and the range is ____

Answer 1

The domain of an exponential function is all real numbers, and the range depends on the value of the base.

Step-by-step explanation:

An exponential function with base b is defined by the function f(x) = b^x, where b > 0 and b ≠ 1. The domain of this function is all real numbers, (-∞, ∞), because any real number can be raised to a power. The range of this function depends on the value of b. If b > 1, then the range is (0, ∞), which means the function will only produce positive values. If 0 < b < 1, then the range is (0, 1), which means the function will only produce values between 0 and 1, but never reaches 0 or 1.

Answer 2

The exponential function f with base b is defined as f(x) = b^x, with b > 0 and b≠1. Its domain is all real numbers, and its range is all positive real numbers.

Step-by-step explanation:

The exponential function f with base b is defined by the equation f(x) = bx, where b is greater than 0 and b is not equal to 1. The reason for these restrictions is because if b were 1, the function would be constant, and if b were 0 or negative, the function would either be undefined or not fit the definition of exponential growth.

Using interval notation, the domain of this function is (-∞, +∞) or simply all real numbers, because you can raise a positive base b to any power x. The range of an exponential function is always (0, +∞), which means it includes all positive real numbers. This is because no matter what real number you raise a positive base to, the result can never be zero or negative.

The concept of exponential arithmetic is used widely to express large and small numbers, growth patterns, and many phenomena in various fields such as physics, biology, and finance.