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Chapter 12: Problem 64

What is an exponential function?

Short Answer

Expert verified
An exponential function is a mathematical function which involves a fixed base raised to the power of a variable exponent. It is widely applied in real-life situations such as population growth or interest computation, and its graph has characteristic properties.

Step by step solution

01

Definition

An exponential function is considered a mathematical function where a fixed base, greater than 1, is raised to the power of a variable exponent.
02

Mathematical Representation

The standard form of an exponential function is \( f(x) = a \cdot b^{x} \) where \( a \) can be any real number and \( b \) is a positive real number other than 1, known as the base.
03

Real-life applications

Exponential functions are used to model a variety of real-life situations such as population growth, radioactive decay, interest computation in finance, to name a few.
04

Key properties

1. The graph of an exponential function always has a horizontal asymptote (a line that a curve approaches, as it heads towards infinity), which is the x-axis, i.e., \( y = 0 \).\n 2. The function is always above the x-axis (positive) but can never touch or cross it.\n 3. The function is always increasing (for \( b > 1 \)) or decreasing (for \( 0 < b < 1 \)).

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Most popular questions from this chapter

Solve each equation. $$\log _{2}(x-6)+\log _{2}(x-4)-\log _{2} x=2$$

Explain the differences between solving \(\log _{3}(x-1)=4\) and \(\log _{3}(x-1)=\log _{3} 4\).

Write as a single term that does not contain a logarithm: $$e^{\ln 8 x^{5}-\ln 2 x^{2}}$$

Explain how to find the domain of a logarithmic function.

What is a logarithmic equation?
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