Problem 77
Question
What is an exponential function?
Step-by-Step Solution
Verified Answer
An exponential function is a mathematical function of the form \(f(x) = b^{x}\), where \(b\) is a positive real number not equal to 1.
1Step 1: Definition of Exponential Function
An exponential function is a mathematical function of the form \(f(x) = b^{x}\), where \(b\) is a positive real number, and \(b \neq 1\). \(b\) is referred to as the 'base' and \(x\) is the exponent.
2Step 2: Properties of Exponential Functions
1. The base \(b\) is always positive and not equal to 1. \n2. The function is increasing if \(b > 1\) and decreasing if \(01\) and \( \lim_{x \to \infty} b^{x} = 0 \) if \(0
3Step 3: Visualizing Exponential Functions
In a graphical representation, an exponential function is a curved line that gets nearer to the x-axis but never touches it as \(x\) decreases, and rises rapidly as \(x\) increases.\nThe curve passes through the point (0,1) representing the y-intercept.
Other exercises in this chapter
Problem 77
Solve each logarithmic equation. Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer.
View solution Problem 77
In Exercises 75–80, find the domain of each logarithmic function. $$ f(x)=\log (2-x) $$
View solution Problem 78
The exponential growth models describe the population of the indicated country, \(A,\) in millions, t years after 2006 . $$ \begin{array}{ll} {\text { Canada }}
View solution Problem 78
Use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. $$ \log _{\pi} 400 $$
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