Problem 77
Question
What is an exponential function?
Step-by-Step Solution
Verified Answer
An exponential function is a mathematical function of the form \(f(x) = a \cdot b^{x}\), where 'a' and 'b' are constants, 'b' is a positive real number, and 'x' is any real number. The graph of this function is always above or on the x-axis. An example is \(f(x) = 2^{x}\), where the value doubles for each unit increase in 'x'.
1Step 1: Definition
An exponential function is a mathematical function of the form \(f(x) = a \cdot b^{x}\), where 'a' and 'b' are constants, 'b' is a positive real number, and 'x' is any real number.
2Step 2: Properties of Exponential Function
1. The base 'b' is a positive real number other than 1.\n2. The graph of an exponential function is always above the x-axis but can approach it as an asymptote.\n3. For a function where \(b > 1\), the slope of the graph will increase as 'x' increases. \n4. For a function where \(0
3Step 3: Example of an Exponential Function
An example of an exponential function is \(f(x) = 2^{x}\). In this function, the value of 'f(x)' doubles for each unit increase in 'x'.
Other exercises in this chapter
Problem 77
Find the domain of each logarithmic function. $$ f(x)=\log (2-x) $$
View solution Problem 77
Solve each logarithmic equation in Exercises \(49-92 .\) Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions.
View solution Problem 78
Find the domain of each logarithmic function. $$ f(x)=\log (7-x) $$
View solution Problem 78
The exponential growth models describe the population of the indicated country, \(A,\) in millions, \(t\) years after 2006 $$\begin{aligned}&\text { Carada } \q
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