2.3 Exponential functions
Write the general form of an exponential function.
What does the initial value a represent in the general form of an exponential function?
What is the difference between exponential growth and exponential decay in an exponential function?
What is the domain of an exponential function?
Describe the graph of an exponential function. How does it change based on the values of a and b?
Given the exponential function f (x) = 2(3)^x, determine whether it demonstrates exponential growth or decay.
Given the function g (x) = 5(0.5)^x + 3:
Identify the transformation that has been applied to the base function.
Describe the effect of the transfomation from part (a) on the graph.
Consider an exponential function f (x) = a(b)^x where a \gt 0 and 0 \lt b \lt 1. Describe the behavior of the function as x approaches positive infinity and negative infinity.
Use the given values to write an exponential function in the form f(x) = ab^x. Initial value a = 3, base b = 2.
Which of the following functions is an exponential function? Justify your answer.
f(x) = 2x^3
g(x) = 3^x
h(x) = \dfrac{1}{x}
k(x) = x^2
Given the function g(x) = f(x) + 5, where f(x) is an exponential function, describe the transformation that g(x) represents.
For the function f(x) = 2^x, calculate the following limits:
\text{lim}_{(x\to \infty)} 2^x
\text{lim}_{(x \to -\infty)} 2^x
If the values of the additive transformation function g (x) = f (x) + k are proportional over equal-length input-value intervals, how can you determine whether function f is exponential?
Given the exponential function f (x) = 2(3)^x.
Describe the long-term behavior of the function as x approaches positive and negative infinity.
Using the answer in part (a), explain why exponential functions do not have extrema except on a closed interval.
Given an exponential function in the form f (x) = ab^x, if b is increased while a remains constant, how will the graph of the function be affected? Support your answer with an example.