Exponential Functions: Properties, Graphs, and Equations

Exponential Functions

Definition and Basic Properties

An exponential function is a function of the form $f(x) = b^x$, where $x$ is any real number and $b$ is a constant such that $b > 0$ and $b \neq 1$. The constant $b$ is called the base of the exponential function.

• Domain: $(-\infty, \infty)$

• Range: $(0, \infty)$

• Intercept: The graph intersects the y-axis at $(0, 1)$ because $b^0 = 1$.

• Horizontal Asymptote: The line $y = 0$ is a horizontal asymptote.

• One-to-one: The function $f(x) = b^x$ is one-to-one.

Graphical Behavior:

• If $b > 1$, the graph increases rapidly as $x$ increases.

• If $0 < b < 1$, the graph decreases rapidly as $x$ increases.

Example: $f(x) = 2^x$ is an increasing exponential function; $f(x) = (1/2)^x$ is a decreasing exponential function.

Characteristics of Exponential Functions

• The graph of $f(x) = b^x$ passes through $(0, 1)$.

• For $b > 1$, the graph rises to the right; for $0 < b < 1$, the graph falls to the right.

• The function never touches the x-axis (horizontal asymptote at $y = 0$).

Example: The graph of $f(x) = 3^x$ passes through $(0, 1)$ and increases as $x$ increases.

The Natural Exponential Function

Definition and Properties

The natural exponential function is $f(x) = e^x$, where $e$ is an irrational number defined as the value of the expression $\left(1 + \frac{1}{n}\right)^n$ as $n$ approaches infinity. The value of $e$ is approximately 2.718281.

• Domain: $(-\infty, \infty)$

• Range: $(0, \infty)$

• Intercept: The graph intersects the y-axis at $(0, 1)$.

• Horizontal Asymptote: $y = 0$

Comparison: The graph of $f(x) = e^x$ lies between the graphs of $f(x) = 2^x$ and $f(x) = 3^x$ when plotted on the same coordinate system.

Table: Values of $\left(1 + \frac{1}{n}\right)^n$ for Increasing $n$

n	$\left(1 + \frac{1}{n}\right)^n$
1	2.0000
2	2.2500
5	2.4883
10	2.5937
100	2.7048
1000	2.7169
10000	2.7181

Transformations of Exponential Functions

Vertical and Horizontal Shifts

Transformations can be used to sketch the graphs of exponential functions. For example, the graph of $f(x) = b^x - 1$ is obtained by shifting the graph of $f(x) = b^x$ down one unit.

• The graph of $f(x) = b^{x-h} + k$ is obtained by shifting $f(x) = b^x$ horizontally by $h$ units and vertically by $k$ units.

• Horizontal asymptote is shifted to $y = k$.

Example: $f(x) = 2^{x} - 1$ has a horizontal asymptote at $y = -1$.

Properties of Exponents

Exponent Rules

• Product Rule: $a^m a^n = a^{m+n}$

• Power of a Power Rule: $(a^m)^n = a^{mn}$

• Power of a Product Rule: $(ab)^n = a^n b^n$

• Quotient Rule: $\frac{a^m}{a^n} = a^{m-n}$, $a \neq 0$

• Zero Exponent Rule: $a^0 = 1$, $a \neq 0$

Negative Exponents: $a^{-n} = \frac{1}{a^n}$, $a \neq 0$

Rational Exponents: $a^{m/n} = \sqrt[n]{a^m}$, $a > 0$

Example: $8^{2/3} = (\sqrt[3]{8})^2 = 2^2 = 4$

Solving Exponential Equations

Method of Relating the Bases

To solve equations of the form $a^{f(x)} = a^{g(x)}$, set the exponents equal: $f(x) = g(x)$.

• If the bases are the same, equate the exponents.

• If the bases are different, rewrite one or both sides using a common base if possible.

Example: Solve $2^{x} = 8$. Rewrite $8$as $2^3$, so $2^x = 2^3$ implies $x = 3$.

Additional info: Not all exponential equations can be solved by relating the bases; other methods (such as logarithms) may be required.