4.1 Exponential Functions

4.1.1 Learning Objectives

By the end of this section, you should be able to:

evaluate exponential functions;
distinguish exponential growth from exponential decay;
graph transformed exponential functions;
interpret the natural base $e$; and
calculate compound and continuously compounded interest.

4.1.2 Exponential Functions and Their Graphs

An exponential function has the form

\[ f(x)=a^x, \]

where $a>0$ and $a\ne1$. Its domain is all real numbers, its range is $(0,\infty)$, its $y$-intercept is $(0,1)$, and $y=0$ is a horizontal asymptote.

If $a>1$, the function increases and represents growth. If $0<a<1$, it decreases and represents decay.

Example 4.1 Evaluating an Exponential Function

For $f(x)=3^x$, find $f(-2)$, $f(0)$, and $f(3)$.

Solution

\[ f(-2)=\frac1{3^2}=\frac19,\qquad f(0)=1,\qquad f(3)=27. \]

\[ \boxed{f(-2)=\frac19,\quad f(0)=1,\quad f(3)=27} \]

Exercise 4.1 For $g(x)=(1/2)^x$, find $g(-1)$, $g(0)$, and $g(3)$.

4.1.3 Transformations of Exponential Functions

For

\[ g(x)=ca^{x-h}+k, \]

$h$ shifts the graph horizontally, $k$ shifts it vertically, and $c$ creates a vertical stretch or compression. A negative $c$ reflects the graph across the $x$-axis. The horizontal asymptote becomes $y=k$.

Example 4.2 Describing an Exponential Transformation

Describe $g(x)=-2^{x-3}+4$ as a transformation of $f(x)=2^x$.

Solution

The graph shifts right $3$, reflects across the $x$-axis, and shifts up $4$. Its horizontal asymptote is $y=4$.

\[ \boxed{\text{Right }3,\quad \text{reflect across the }x\text{-axis},\quad \text{up }4} \]

Exercise 4.2 State the horizontal asymptote of $g(x)=5(3^{x+1})-2$.

4.1.4 Exponential and Power Functions

A power function such as $x^4$ has a constant exponent and a variable base. An exponential function such as $4^x$ has a constant base and a variable exponent. Exponential growth eventually exceeds the growth of every fixed-degree power function.

Example 4.3 Comparing Two Types of Growth

Compare $f(x)=x^3$ and $g(x)=3^x$ at $x=5$ and $x=10$.

Solution

\[ f(5)=125,\qquad g(5)=243, \]

and

\[ f(10)=1000,\qquad g(10)=59049. \]

The exponential function is already larger at both inputs, and the difference grows rapidly.

\[ \boxed{g(5)>f(5)\quad\text{and}\quad g(10)>f(10)} \]

Exercise 4.3 Which is exponential: $f(x)=x^6$ or $g(x)=6^x$?

4.1.5 The Natural Exponential Function

The number

\[ e\approx2.71828 \]

is the base of the natural exponential function $f(x)=e^x$. It appears naturally in continuous growth and decay models. Its graph has the same basic features as any exponential growth function.

Example 4.4 Evaluating a Natural Exponential

Evaluate $f(x)=4e^{0.3x}$ at $x=5$.

Solution

\[ f(5)=4e^{1.5}\approx4(4.48169)\approx17.93. \]

\[ \boxed{f(5)\approx17.93} \]

Exercise 4.4 Evaluate $3e^{-0.4(2)}$ to two decimal places.

4.1.6 Compound Interest

If a principal $P$ earns annual interest rate $r$, compounded $n$ times per year for $t$ years, then

\[ A=P\left(1+\frac rn\right)^{nt}. \]

For continuous compounding,

\[ A=Pe^{rt}. \]

Rates must be written as decimals, and $A$ and $P$ use the same currency units.

Example 4.5 Comparing Compounding Methods

Find the value of a $\$2500$ investment after $6$ years at $4.8\%$ per year, compounded monthly.

Solution

\[ \begin{aligned} A&=2500\left(1+\frac{0.048}{12}\right)^{12(6)}\\ &\approx3332.48. \end{aligned} \]

\[ \boxed{A\approx\$3332.48} \]

Exercise 4.5 Find the value of $\$1800$ after $5$ years at $3.5\%$, compounded continuously.

4.1.7 Conceptual Takeaways

Exponential functions have the variable in the exponent.
The base determines whether the function grows or decays.
Exponential outputs remain positive before vertical shifts or reflections.
The base $e$ is especially useful for continuous change.
Compounding frequency affects accumulated value.

4.1.8 Skills You Should Be Able to Do

Evaluate exponential expressions.
Identify growth, decay, domain, range, and asymptotes.
Graph exponential transformations with Desmos.
Use $e$ in continuous-change calculations.
Calculate compound interest.

4.1.9 Practice Problems with Solutions

Evaluate $f(x)=4^x$ at $x=-2$.

Show Solution

\[ f(-2)=4^{-2}=\frac1{16}. \]

\[ \boxed{\frac1{16}} \]
Determine whether $f(x)=(0.7)^x$ represents growth or decay.

Show Solution

Since $0<0.7<1$, the function decreases as $x$ increases. It represents $\boxed{\text{exponential decay}}$.
State the domain, range, and horizontal asymptote of $g(x)=3^{x-2}+5$.

Show Solution

The domain is all real numbers. Since $3^{x-2}>0$, $g(x)>5$.

\[ \boxed{\text{Domain }(-\infty,\infty),\quad \text{range }(5,\infty),\quad y=5} \]
Use Desmos to graph $f(x)=2^x$ and $g(x)=2^{x+3}-1$. Describe the transformation.

Show Solution

Replacing $x$ with $x+3$ shifts left $3$, and subtracting $1$ shifts down $1$.

\[ \boxed{\text{Left }3\text{ and down }1} \]
Find the value of $\$4000$ after $8$ years at $5.2\%$, compounded quarterly.

Show Solution

\[ A=4000\left(1+\frac{0.052}{4}\right)^{32}\approx6047.31. \]

\[ \boxed{\$6047.31} \]
Compare annual and continuous compounding for $\$3000$ invested at $4\%$ for $10$ years.

Show Solution

Annual compounding gives

\[ A=3000(1.04)^{10}\approx4440.73. \]

Continuous compounding gives

\[ A=3000e^{0.4}\approx4475.47. \]

Therefore, continuous compounding produces about $\$34.74$ more.

\[ \boxed{\$4440.73\text{ annually};\quad \$4475.47\text{ continuously}} \]