Introduction to Exponential Functions

Exponential Functions

Introduction to Exponential Functions

Exponential functions are a fundamental concept in precalculus, contrasting with polynomial functions by having the variable in the exponent rather than the base. Understanding the structure and properties of exponential functions is essential for modeling growth and decay in various real-world contexts.

• Polynomial functions have a variable base with a number exponent.

• Exponential functions have a constant base and a variable exponent.

Polynomial Function	Exponential Function
$f(x) = x^n$ Base: $x$ Power: $n$	$f(x) = a^x$ Base: $a$ Power: $x$

Key Properties of Exponential Functions

• Base ($a$): A positive real number, $a > 0$, $a \neq 1$.

• Exponent ($x$): The variable in the function.

• General form: $f(x) = a^x$

Examples: Identifying Exponential Functions

• Example 1: $f(x) = 2^x$ is an exponential function (base $2$, power$x$).

• Example 2: $g(x) = x^2$ is not an exponential function (variable base, constant exponent).

• Example 3: $h(x) = 3^{x+1}$ is an exponential function (base $3$, power$x+1$).

Evaluating Exponential Functions

To evaluate an exponential function for a specific value of $x$, substitute the value into the exponent and calculate the result.

• Example: Evaluate $f(x) = 2^x$ at $x = 3$:

$f(3) = 2^3 = 8$

Practice Problems

• Determine if each function is an exponential function. If so, identify the power and base, then evaluate for $x = 4$.

Function	Exponential?	Base	Power	Evaluate at $x=4$
$f(x) = 4 \cdot 2^x$	Yes	2	$x$	$4 \cdot 2^4 = 4 \cdot 16 = 64$
$g(x) = 3x \cdot 5^x$	Yes	5	$x$	$3 \cdot 4 \cdot 5^4 = 12 \cdot 625 = 7500$
$h(x) = (\frac{1}{2})^x$	Yes	$\frac{1}{2}$	$x$	$(\frac{1}{2})^4 = \frac{1}{16}$

Summary

• Exponential functions have the form $f(x) = a^x$ with $a > 0$, $a \neq 1$.

• The variable is in the exponent, not the base.

• They are used to model rapid growth or decay in many applications.