Exponential Functions: Definitions, Graphs, and Equations

Exponential Functions

Definition of Exponential Functions

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: $(0, \infty)$

Characteristics of Exponential Functions

For $b > 1$, the exponential function $f(x) = b^x$ is increasing. For $0 < b < 1$, $f(x) = b^x$ is decreasing. The graph of $f(x) = b^x$ has one of the following shapes depending on the value of $b$:

• Intercept: The graph always passes through $(0, 1)$ because $b^0 = 1$.

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

• Behavior:

  • If $b > 1$, as $x \to \infty$, $f(x) \to \infty$; as $x \to -\infty$, $f(x) \to 0$.

  • If $0 < b < 1$, as $x \to \infty$, $f(x) \to 0$; as $x \to -\infty$, $f(x) \to \infty$.

• One-to-one: Exponential functions are one-to-one, meaning each $x$ value corresponds to a unique $f(x)$ value.

Example:

For $f(x) = 2^x$:

• Domain: $(-\infty, \infty)$

• Range: $(0, \infty)$

• Intercept: $(0, 1)$

• Horizontal asymptote: $y = 0$

The Number $e$ and the Natural Exponential Function

The number $e$ is an irrational constant defined as the value of $(1 + \frac{1}{n})^n$ as $n$ approaches infinity. Its approximate value is $e \approx 2.7182818$.

n	$(1 + \frac{1}{n})^n$
1	2.000000
2	2.250000
5	2.488320
10	2.593742
100	2.704814
1000	2.716924
10000	2.718146
100000	2.718269
1000000	2.718280

The natural exponential function is $f(x) = e^x$.

• Domain: $(-\infty, \infty)$

• Range: $(0, \infty)$

• Intercept: $(0, 1)$

• Horizontal asymptote: $y = 0$

Example:

The graph of $f(x) = e^x$ passes through $(0, 1)$ and approaches $y = 0$ as $x \to -\infty$.

Transformations of Exponential Functions

Exponential functions can be transformed by shifting, stretching, or reflecting their graphs. For example, $f(x) = 3^x - 1$ is obtained by shifting the graph of $f(x) = 3^x$ down one unit.

• Vertical Shift: $f(x) = b^x + k$ shifts the graph up ($k > 0$) or down ($k < 0$).

• Horizontal Shift: $f(x) = b^{x-h}$ shifts the graph right ($h > 0$) or left ($h < 0$).

• Horizontal Asymptote: The line $y = k$ is the new horizontal asymptote after a vertical shift.

Example:

The graph of $f(x) = 3^x - 1$ passes through $(-1, -2)$, $(0, 0)$, and $(1, 2)$, with horizontal asymptote $y = -1$.

Solving Exponential Equations by Relating the Bases

To solve equations of the form $b^x = b^y$, use the Method of Relating the Bases:

• If the bases are the same, set the exponents equal: $x = y$.

• Rewrite the equation so both sides have the same base if possible.

Example:

Solve $2^{x+1} = 2^5$:

• Set exponents equal: $x+1 = 5$

• Solution: $x = 4$

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