Calculus I > Precalculus Review
Review Objective
Basic Concepts of Functions
Defining a Function
Intervals
Implied Domain
Even and Odd Functions
Composite functions
Inverse Functions
Polynomial Functions
Linear Functions
Quadratic Functions
Polynomial Function
Rational Functions
Rational Functions
Vertical and Horizontal Asymptote
Exponential and Logarithm
Rational and Irrational Exponents
Exponential Functions
Logarithmic Functions
Laws of Logarithms
Trigonometric functions
Radian and right triangles
Trigonometric functions
Trigonometric functions on a coordinate system
Sine and cosine function
Tangent function
Trigonometric additions and subtractions
Trigonometric formulas
Inverse sine function
Inverse cosine function
Inverse tangent function

Exponential Functions

A function $ f$ is called an exponential function if

$\displaystyle f(x) = a^x . $

When $ a > 1$, $ f$ is an increasing function and $ f(x) \to 0$ as $ x \to -\infty$. When $ 0 < a < 1$, $ f$ is a decreasing function and $ f(x) \to 0$ as $ x \to \infty$. In either case, $ f$ is a one-to-one function, and has the horizontal asymptote $ y = 0$. The domain $ D$ and the range $ R$ of $ f$ are given by $ D = (-\infty, \infty)$ and $ R = (0, \infty)$, respectively.

1. $ a > 1$ 2. $ 0 < a < 1$
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Property of one-to-one functions. If $ f$ is a one-to-one function, $ f(x_1) = f(x_2)$ implies $ x_1 = x_2$. In particular, if $ f(x) = a^x$ with positive real value $ a \neq 1$, $ a^{x_1} = a^{x_2}$ implies $ x_1 = x_2$.

Natural exponential function. A function $ f$ is called the natural exponential function if

$\displaystyle f(x) = e^x $

where $ e \approx 2.71828\ldots$ is the number given by (1).

Exponential growth and decay. Suppose that the value $ q(t)$ at time $ t$ is expressed as

$\displaystyle q(t) = a e^{r t} $

with initial value $ a > 0$ at time $ t = 0$. If $ r > 0$, we say that the value $ q(t)$ “increases exponentially” and $ r$ is often called the “rate of growth.” If $ r < 0$, we say that the value $ q(t)$ “decreases exponentially” and $ r$ is often called the “rate of decay.”

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