a Function f is defined as f(x)=3*2^x
a)is f increasing or decreasing?
b)write equation for the horizontal asymptote
c)describe the end behavior of f
To determine whether the function f(x) = 3 * 2^x is increasing or decreasing, we need to examine the growth rate of the function.
a) To find out if f(x) is increasing or decreasing, we can examine the sign of its derivative. Differentiating the function with respect to x, we get:
f'(x) = 3 * (2^x) * ln(2)
Since ln(2) is positive, we only need to consider the sign of (2^x).
(2^x) represents an exponential growth function, and for all positive values of x, 2^x will also be positive. Therefore, f'(x) will be positive for all x.
Since f'(x) is always positive, we can conclude that f(x) is an increasing function.
b) Horizontal asymptotes represent values that the function approaches as x approaches infinity or negative infinity. To find the horizontal asymptote, we need to examine the end behavior of f(x).
As x approaches negative infinity (-∞), 2^x approaches 0, and since 2^x is in the numerator, f(x) approaches 0/3 = 0. Therefore, the equation for the horizontal asymptote is y = 0.
c) The end behavior of f(x) can be described as follows:
- As x approaches negative infinity, f(x) approaches 0.
- As x approaches infinity, f(x) grows without bound, since 2^x increases rapidly as x gets larger.