﻿ Exponential Functions: Review and Additional Properties

# EXPONENTIAL FUNCTIONS: REVIEW AND ADDITIONAL PROPERTIES

by Dr. Carol JVF Burns (website creator)
Follow along with the highlighted text while you listen!
• PRACTICE (online exercises and printable worksheets)

An exponential function is a function of the form $\,f(x) = b^x\,$ for $\,b > 0\,$ and $\,b\ne 1\,$.
Notice that the variable is in the exponent.
For example, $\,y = 2^x\,$, $\,y = (\frac 12)^x\,$, and $\,y = \text{e}^x\,$ are exponential functions.

Exponential functions and their graphs were introduced in the Algebra II curriculum:

Additional higher-level information that is important for Precalculus is presented below.

## Increasing/Decreasing Properties of Exponential Functions

 When $\,b > 1\,$, $\,f(x) = b^x\,$ is an increasing function. That is, for all real numbers $\,x\,$ and $\,y\,$: $$\cssId{s13}{x < y} \ \ \cssId{s14}{\Rightarrow}\ \ \cssId{s15}{b^x < b^y}$$ It is clear from the graph that the other direction is also true: $$\cssId{s17}{b^x < b^y} \ \ \cssId{s18}{\Rightarrow}\ \ \cssId{s19}{x < y}$$ Together, we have: $$\cssId{s21}{x < y}\ \ \cssId{s22}{\text{is equivalent to}}\ \ \cssId{s23}{b^x < b^y} \tag{1}$$ Notice that when the base is greater than one, the inequality symbols that compare the inputs ($\,x\,$ and $\,y\,$) and their corresponding outputs ($\,b^x\,$ and $\,b^y\,$) have the same direction: $$\cssId{s26}{x \ {\bf\color{red}{\lt }}\ y}\ \ \cssId{s27}{\text{is equivalent to}}\ \ \cssId{s28}{b^x\ {\bf\color{red}{\lt }}\ b^y} \tag{1a}$$ This equivalence can be alternatively stated as: $$\cssId{s30}{x \ {\bf\color{red}{>}}\ y}\ \ \cssId{s31}{\text{is equivalent to}}\ \ \cssId{s32}{b^x\ {\bf\color{red}{>}}\ b^y} \tag{1b}$$ $y = b^x\,$, for $\,b > 1\,$ an increasing exponential function $x < y \ \ \iff\ \ b^x < b^y$ When $\,0 < b < 1\,$, $\,f(x) = b^x\,$ is a decreasing function. That is, for all real numbers $\,x\,$ and $\,y\,$: $$\cssId{s39}{x < y \ \ \Rightarrow\ \ b^x > b^y}$$ It is clear from the graph that the other direction is also true: $$\cssId{s41}{b^x > b^y \ \ \Rightarrow\ \ x < y}$$ Together, we have: $$\cssId{s43}{x < y\ \ \text{is equivalent to}\ \ b^x > b^y} \tag{2}$$ Notice that when the base is between zero and one, the inequality symbols that compare the inputs ($\,x\,$ and $\,y\,$) and their corresponding outputs ($\,b^x\,$ and $\,b^y\,$) have different directions: $$\cssId{s46}{x \ {\bf\color{red}{\lt}}\ y\ \ \text{is equivalent to}\ \ b^x\ {\bf\color{red}{\gt}}\ b^y} \tag{2a}$$ This equivalence can be alternatively stated as: $$\cssId{s48}{x \ {\bf\color{red}{\gt}}\ y\ \ \text{is equivalent to}\ \ b^x\ {\bf\color{red}{\lt}}\ b^y} \tag{2b}$$ $y = b^x\,$, for $\,0 < b < 1\,$ a decreasing exponential function $x < y \ \ \iff\ \ b^x > b^y$

## Exponential Functions are One-to-One

The graphs of exponential functions pass both vertical and horizontal lines tests, so they are one-to-one functions.
Thus: $$\cssId{s56}{x = y \ \ \text{is equivalent to}\ \ b^x = b^y} \tag{3}$$ Consequently, exponential functions have inverses.
In a future section, we'll see that the class of logarithmic functions provide inverses to the class of exponential functions.

## Solving Inequalities involving Exponential Functions

Equivalences (1), (2), and (3) can be used to easily solve certain types of mathematical sentences, as illustrated in the following examples.

 Solve: $2^{3x-1} \lt 2^{5x}$ This is an inequality of the form $$\, \cssId{s64}{b^{\text{stuff1}} < b^{\text{stuff2}}}\,$$ where the base, $\,b\,$, is $\,2\,$. Notice that the exponential functions on both sides use the same base. In this example, the base is greater than one, so we'll use equivalence (1): $$\cssId{s68}{x < y \ \ \iff\ \ b^x < b^y}$$ SOLUTION: \begin{alignat}{2} \cssId{s70}{2^{3x-1}} \ & \cssId{s71}{\lt} \ \cssId{s72}{2^{5x}} \qquad&&\cssId{s73}{\text{original inequality}}\cr\cr \cssId{s74}{3x-1} & \cssId{s75}{\lt} \cssId{s76}{5x} && \cssId{s77}{\text{use (1); inequality symbol doesn't change}}\cr\cr \cssId{s78}{-2x} & \cssId{s79}{\lt} \cssId{s80}{1} &&\cssId{s81}{\text{addition property of inequality}}\cr\cr \cssId{s82}{x} & \cssId{s83}{\gt} \cssId{s84}{-\frac 12} && \cssId{s85}{\text{divide by a negative #; reverse inequality}} \end{alignat} blue curve:   $y = 2^{3x - 1}$ red curve:   $y = 2^{5x}$ blue curve lies below red curve for $\,x > -\frac 12$ Solve: $\displaystyle\frac 1{3^{x^2-1}} > 1$ This inequality will be solved in two different ways. As long as you use correct tools in a correct way, there are often different ways you can proceed. SOLUTION #1: \begin{alignat}{2} \cssId{s96}{\frac 1{3^{x^2-1}}} \ & \cssId{s97}{\gt}\ \cssId{s98}{1} \qquad&&\cssId{s99}{\text{original inequality}}\cr\cr \cssId{s100}{\left(\frac 13\right)^{x^2-1}}\ & \cssId{s101}{\gt}\ \cssId{s102}{\left(\frac 13\right)^0} \ \ &&\cssId{s103}{\text{rename both sides using exponent laws}}\cr\cr \cssId{s104}{x^2 - 1}\ & \cssId{s105}{\lt} \cssId{s106}{0}&&\cssId{s107}{\text{use (2); inequality changes direction}}\cr\cr \cssId{s108}{x^2} & \cssId{s109}{\lt} \cssId{s110}{1} && \cssId{s111}{\text{addition property of inequality}}\cr\cr \cssId{s112}{-1} \cssId{s113}{\lt} &\ \cssId{s114}{x} \cssId{s115}{\lt} \cssId{s116}{1} && \cssId{s117}{\text{inspection (knowledge of x^2 curve)}} \end{alignat} SOLUTION #2: \begin{alignat}{2} \cssId{s119}{\frac 1{3^{x^2-1}}} \ & \cssId{s120}{\gt} \ \cssId{s121}{1} \qquad&&\cssId{s122}{\text{original inequality}}\cr\cr \cssId{s123}{1}\ & \cssId{s124}{\gt}\ \cssId{s125}{3^{x^2-1}} \ \ \ \ &&\cssId{s126}{\text{multiply both sides by 3^{x^2-1}\overset{\text{always}}{\gt}0}}\cr\cr \cssId{s127}{3^0} \ & \cssId{s128}{\gt}\ \cssId{s129}{3^{x^2-1}}&&\cssId{s130}{\text{rename: 1 = 3^0}}\cr\cr \cssId{s131}{0}\ \ & \cssId{s132}{\gt}\ \cssId{s133}{x^2 - 1}\ \ &&\cssId{s134}{\text{use (1)}}\cr\cr \cssId{s135}{x^2} \ & \cssId{s136}{\lt}\ \cssId{s137}{1}&&\cssId{s138}{\text{addition property; re-arrange}}\cr\cr \cssId{s139}{-1} \cssId{s140}{\lt} \ &\cssId{s141}{x}\ \cssId{s142}{\lt} \cssId{s143}{1}&&\cssId{s144}{\text{inspection (knowledge of x^2 curve)}} \end{alignat} blue curve:   $\displaystyle y = \frac{1}{3^{x^2-1}}$ red curve:   $y = 1$ blue curve lies above red curve for $\,x\,$ between $\,-1\,$ and $\,1\,$
Master the ideas from this section