Functions, Exponential and Logarithmic Functions in Business Calculus

1. Introduction to Functions

Definition and Representation

Functions are fundamental objects in calculus, representing relationships between two sets where each input has exactly one output.

• Function: A rule that assigns to each element in a set (domain) exactly one element in another set (range).

• Notation: $f(x)$ denotes the value of the function $f$ at input $x$.

• Graph: The set of all points $(x, f(x))$ in the coordinate plane.

Example: The function $f(x) = x^2$ maps each real number $x$ to its square.

2. Types of Functions

Linear, Quadratic, Polynomial, Rational, Exponential, and Logarithmic Functions

Different types of functions are used to model various business and economic phenomena.

• Linear Function: $f(x) = mx + b$

• Quadratic Function: $f(x) = ax^2 + bx + c$

• Polynomial Function: $f(x) = a_nx^n + \ldots + a_1x + a_0$

• Rational Function: $f(x) = \frac{p(x)}{q(x)}$ where $p(x)$ and $q(x)$ are polynomials and $q(x) \neq 0$

• Exponential Function: $f(x) = a^x$ where $a > 0$, $a \neq 1$

• Logarithmic Function: $f(x) = \log_a x$ where $a > 0$, $a \neq 1$

Example: $f(x) = 2^x$ is an exponential function; $f(x) = \log_2 x$ is a logarithmic function.

3. Graphs of Functions

Key Features and Transformations

Understanding the graphs of functions helps in visualizing their behavior and transformations.

• Intercepts: Points where the graph crosses the axes.

• Asymptotes: Lines that the graph approaches but never touches (common in rational, exponential, and logarithmic functions).

• Domain and Range: The set of possible input and output values, respectively.

• Shifts and Reflections: Transformations such as $f(x) + c$ (vertical shift), $f(x + c)$ (horizontal shift), $-f(x)$ (reflection over x-axis).

Example: The graph of $f(x) = e^x$ passes through $(0,1)$ and has a horizontal asymptote at $y=0$.

4. Exponential and Logarithmic Functions

Definitions and Properties

Exponential and logarithmic functions are widely used in business for modeling growth, decay, and financial calculations.

• Exponential Function: $f(x) = a^x$

• Natural Exponential Function: $f(x) = e^x$ where $e \approx 2.718$

• Logarithmic Function: $f(x) = \log_a x$ is the inverse of $a^x$

• Natural Logarithm: $f(x) = \ln x$ is the logarithm with base $e$

Properties of Logarithms:

• $\log_a (xy) = \log_a x + \log_a y$

• $\log_a \left(\frac{x}{y}\right) = \log_a x - \log_a y$

• $\log_a (x^r) = r \log_a x$

• $\log_a a = 1$

• $\log_a 1 = 0$

Change of Base Formula:

• $\log_a x = \frac{\ln x}{\ln a}$

Example: $\log_2 8 = 3$ because $2^3 = 8$.

5. Inverse Functions

Definition and Finding Inverses

An inverse function reverses the effect of the original function, swapping the roles of inputs and outputs.

• Definition: If $f(x)$ is a one-to-one function, its inverse $f^{-1}(x)$ satisfies $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.

• Finding the Inverse: To find $f^{-1}(x)$, solve $y = f(x)$ for $x$ in terms of $y$, then interchange $x$ and $y$.

Example: For $f(x) = 2x + 3$, the inverse is $f^{-1}(x) = \frac{x-3}{2}$.

6. Properties of Exponential and Logarithmic Functions

Key Properties and Graphs

• Exponential Growth and Decay: $y = Ce^{kt}$, where $C$ is the initial value and $k$ is the growth ($k>0$) or decay ($k<0$) rate.

• Graphs: Exponential functions have horizontal asymptotes; logarithmic functions have vertical asymptotes.

Example: The function $y = e^{-x}$ models exponential decay.

7. Summary Table: Properties of Exponential and Logarithmic Functions

Function	Domain	Range	Asymptote	Key Point
$f(x) = a^x$	$(-\infty, \infty)$	$(0, \infty)$	$y = 0$	$(0,1)$
$f(x) = \log_a x$	$(0, \infty)$	$(-\infty, \infty)$	$x = 0$	$(1,0)$

Additional info: This table summarizes the main properties of exponential and logarithmic functions, which are essential for understanding their applications in business calculus.