True or False. If possible kindly explain why so that I can understand it. Thanks much.
1. f(x) = 1^x is not an exponential function.
2. Every exponential function is strictly increasing.
1 false
a^x is exponential, for any positive a.
#2 false, since 1^x is constant.
Most mathematical authorities define an exponential function as
f(x) = a^x, where a is any positive number, a ≠ 1, and x is any real number
so #1 is True
#2, what about y = (1/2)^x
as x increases, y decreases, so False
1. False.
To determine whether the function f(x) = 1^x is an exponential function, we need to consider its form. An exponential function is generally in the form f(x) = a^x, where 'a' is a constant.
In this case, the function is f(x) = 1^x. When we evaluate 1^x, the base 1 raised to any power will always result in 1. Therefore, f(x) = 1^x simplifies to f(x) = 1, which is a constant function.
Since the function f(x) = 1^x does not follow the general form of an exponential function, it is not an exponential function.
2. True.
Exponential functions, by definition, have a characteristic of consistent growth or decay as the input variable increases or decreases. This implies that the function is strictly increasing or strictly decreasing.
For example, if we have the exponential function f(x) = a^x, where 'a' is a positive constant greater than one, as x increases, the value of f(x) also increases. This continuous growth makes exponential functions strictly increasing.
Similarly, if we have the exponential function f(x) = a^x, where 'a' is a positive constant between zero and one, as x increases, the value of f(x) decreases. This continuous decay also makes exponential functions strictly decreasing.
Therefore, every exponential function is indeed strictly increasing or strictly decreasing.