General Exponential Functions (College Board AP® Precalculus): Study Guide

General exponential functions

What is the general form of an exponential function?

• The general form of an exponential function is $f\left(x\right)=a{b}^x$

  • $a$ is the initial value

    • i.e. the output when $x=0$

      • since $f\left(0\right)=a{b}^0=a$

    • $a\ne 0$ (otherwise the function is just the constant zero)

  • $b$ is the base

    • The base must satisfy $b>0$ and $b\ne 1$

• When $\mathit{a}\mathbf{>}\mathbf{0}$ and $\mathit{b}\mathbf{>}\mathbf{1}$, the function demonstrates exponential growth

  • the output values increase as $x$ increases

• When $\mathit{a}\mathbf{>}\mathbf{0}$ and $\mathbf{0}\mathbf{<}\mathit{b}\mathbf{<}\mathbf{1}$, the function demonstrates exponential decay

  • the output values decrease as $x$ increases

• When $\mathit{a}\mathbf{<}\mathbf{0}$, the graph is reflected across the $x$-axis compared to the corresponding positive $a$ case

What is the domain of an exponential function?

• The domain of an exponential function is all real numbers

• When the inputs are natural numbers ($1,2,3,\dots$), the input value specifies

  • the number of factors of the base $b$ to apply to the initial value $a$

  • E.g. for $f\left(x\right)=5·3^x$

    • $f\left(1\right)=5·3$

    • $f\left(2\right)=5·3·3$

    • $f\left(3\right)=5·3·3·3$

    • etc.

  • This connects directly to the idea of geometric sequences as repeated multiplication

• However an exponential function is also defined for non-integer and negative inputs

  • E.g. for $f\left(x\right)=5·3^x$

    • $f(0.5)=5·3^{0.5}=5\sqrt{3}$

    • $f(-1)=5·3^{-1}=\frac{5}{3}$

What key characteristics does an exponential function have?

• The output values of an exponential function in general form are proportional over equal-length input-value intervals; therefore

  • Exponential functions are always increasing or always decreasing

    • They do not change direction

  • Their graphs are always concave up or always concave down

    • The concavity does not change

• As a consequence:

  • Exponential functions do not have extrema (i.e. maximum or minimum values)

    • except on a closed interval

  • Their graphs do not have points of inflection

How can you identify a 'hidden' exponential function?

• Sometimes data does not immediately appear to be exponential, because a constant has been added to the output values

• If the output values of a function $f$ are not proportional over equal-length input-value intervals

  • but the output values of an additive transformation $g\left(x\right)=f\left(x\right)+k$ are proportional over equal-length input-value intervals

  • then $f$ can be modeled by an additive transformation of an exponential function

• E.g. consider the data points $(1,7)$, $(2,9)$, $(3,13)$, $(4,21)$

  • The ratios of successive outputs are $\frac{9}{7}$, $\frac{13}{9}$, $\frac{21}{13}$

    • These are not constant, so the data does not look exponential at first

  • But if you subtract 5 from each output value, you get $(1,2)$, $(2,4)$, $(3,8)$, $(4,16)$

  • Now the ratios are $\frac{4}{2}=2$, $\frac{8}{4}=2$, $\frac{16}{8}=2$

    • I.e. they are constant

  • So the original data can be modeled by $f\left(x\right)=2^x+5$

    • an exponential function with a vertical shift

What are the end behaviors of an exponential function?

• The end behavior of an exponential function $a{b}^x$ depends on the values of $a$ and $b$

  • This is summarised in the table below

  • Note that changing $a>0$ to $a<0$ 'flips' the values of the unbounded limits

The natural base e

What is the natural base e?

• The number $\mathit{e}$ is a special mathematical constant, approximately equal to 2.718

  • Like $\pi$, $e$ is an irrational number

    • Its decimal expansion goes on forever without repeating

    • $e=2.7182818284590452353602874713527...$

• $e$ is often used as the base in exponential functions that model real-world phenomena

  • For example, continuous growth and decay processes in science, finance, and other fields are often expressed using base $e$

• An exponential function with base $e$ is written as $f\left(x\right)=a{e}^x$

  • This follows the same general form $f\left(x\right)=a{b}^x$

    • with $b=e$

• Since $e\approx 2.718>1$, the function $f\left(x\right)=e^x$ (with $a=1>0$) demonstrates exponential growth

• You will encounter the natural base $e$ frequently in modeling contexts