Examples: f(x) = 2e^x, f : x ↦ log x, f⁻¹(x), fg(x), f²(x)

Introduction

Key Concepts

1. Exponential Functions: f(x) = 2e^x

Exponential functions are a class of functions where the variable appears in the exponent. The general form of an exponential function is $f(x) = a e^{bx}$, where:

• a is the initial value or y-intercept.
• e is the base of the natural logarithm, approximately equal to 2.71828.
• b is the growth (if positive) or decay (if negative) rate.

In the example $f(x) = 2e^{x}$:

• Initial Value (a): 2. This means when $x = 0$, $f(0) = 2e^{0} = 2(1) = 2$.
• Growth Rate (b): 1, indicating exponential growth.

**Graphical Representation:** The graph of an exponential growth function rises rapidly as $x$ increases. It has a horizontal asymptote at $y = 0$.

**Applications:** Exponential functions model various real-life phenomena, including population growth, radioactive decay, and interest calculations in finance.

**Example Problem:**

1. Calculate $f(2)$ for the function $f(x) = 2e^{x}$.

**Solution:**

2. Logarithmic Functions: f : x ↦ log x

A logarithmic function is the inverse of an exponential function. The standard form is $f(x) = \log_{b}(x)$, where:

• b is the base of the logarithm.

In the given example, $f : x ↦ \log x$, it is implied that the base is 10 unless otherwise specified. So, $f(x) = \log_{10}(x)$.

**Key Properties:**

• Domain: $x > 0$
• Range: All real numbers.
• Intercept: $(1, 0)$
• Asymptote: Vertical asymptote at $x = 0$.

**Graphical Representation:** The logarithmic function increases slowly as $x$ increases. It passes through the point $(1, 0)$ and approaches the vertical asymptote $x = 0$.

**Applications:** Logarithmic functions are used in measuring sound intensity (decibels), in the Richter scale for earthquake magnitude, and in calculating pH levels in chemistry.

**Example Problem:**

1. Find the value of $x$ such that $f(x) = \log_{10}(x) = 3$.

**Solution:**

3. Inverse Functions: f⁻¹(x)

The inverse of a function reverses the roles of inputs and outputs. If a function $f$ takes an input $x$ and produces an output $y$, then its inverse function $f^{-1}$ takes $y$ as input and returns $x$.

**Definition:** $$ f^{-1}(f(x)) = x \quad \text{and} \quad f(f^{-1}(x)) = x $$

**Finding the Inverse:**

1. Start with the equation $y = f(x)$.
2. Swap $x$ and $y$: $x = f(y)$.
3. Solve for $y$ in terms of $x$: $y = f^{-1}(x)$.

**Example: Inverse of $f(x) = 2e^{x}$**

1. Start with $y = 2e^{x}$.
2. Swap $x$ and $y$: $x = 2e^{y}$.
3. Solve for $y$:
$$ \frac{x}{2} = e^{y} \\ \ln\left(\frac{x}{2}\right) = y \\ y = \ln\left(\frac{x}{2}\right) $$
4. Thus, $f^{-1}(x) = \ln\left(\frac{x}{2}\right)$.

**Graphical Interpretation:** The graph of an inverse function is the reflection of the original function across the line $y = x$.

**Applications:** Inverse functions are used in various fields such as cryptography, where encoding and decoding processes rely on inverse operations.

**Example Problem:**

1. Find the inverse of $f(x) = \log_{10}(x)$.

**Solution:**

1. Start with $y = \log_{10}(x)$.
2. Swap $x$ and $y$: $x = \log_{10}(y)$.
3. Solve for $y$:
$$ 10^{x} = y \\ y = 10^{x} $$
4. Thus, $f^{-1}(x) = 10^{x}$.

4. Product of Functions: fg(x)

The product of two functions $f$ and $g$, denoted as $fg(x)$ or $f(x)g(x)$, is a function obtained by multiplying the outputs of $f$ and $g$ for the same input $x$.

**Definition:** $$ (fg)(x) = f(x) \cdot g(x) $$

**Example:**

Let $f(x) = 2e^{x}$ and $g(x) = \log_{10}(x)$.

Then, $$ (fg)(x) = 2e^{x} \cdot \log_{10}(x) $$

**Domain Consideration:** The domain of $fg(x)$ is the intersection of the domains of $f(x)$ and $g(x)$. Since $f(x) = 2e^{x}$ is defined for all real numbers and $g(x) = \log_{10}(x)$ is defined for $x > 0$, the domain of $fg(x)$ is $x > 0$.

**Graphical Representation:** The graph of $fg(x)$ combines the increasing nature of $e^{x}$ with the logarithmic growth of $\log_{10}(x)$. The behavior of the product function depends on the interplay between these two components.

**Applications:** Product functions are used in areas like physics for calculating work done ($Work = Force \times Distance$) and in economics for determining total revenue ($Revenue = Price \times Quantity$).

**Example Problem:**

1. Given $f(x) = 2e^{x}$ and $g(x) = \log_{10}(x)$, find $fg(1)$.

**Solution:**

5. Power Functions: f²(x)

Power functions involve raising a function to a positive integer exponent. The notation $f^{n}(x)$ represents the function $f(x)$ raised to the power of $n$.

**Definition:** $$ f^{2}(x) = (f(x))^{2} $$

**Example:**

Given $f(x) = 2e^{x}$, then: $$ f^{2}(x) = (2e^{x})^{2} = 4e^{2x} $$

**Properties:**

• Non-Negativity: Since squares of real numbers are non-negative, $f^{2}(x) \geq 0$ for all $x$.
• Growth Rate: The growth rate is faster compared to the original function if the exponent is greater than 1.

**Graphical Representation:** The graph of $f^{2}(x)$ is a steeper curve compared to $f(x)$ due to the squared term, especially noticeable for exponential functions.

**Applications:** Power functions are prevalent in physics, such as calculating kinetic energy ($KE = \frac{1}{2}mv^{2}$), and in statistics for variance calculations.

**Example Problem:**

1. Find $f^{2}(x)$ if $f(x) = \log_{10}(x)$.

**Solution:**

Advanced Concepts

1. Inverse Functions in Depth

Inverse functions play a crucial role in various mathematical operations, enabling the reversal of processes modeled by the original function. Understanding inverse functions is essential for solving equations, performing transformations, and analyzing function behavior.

**Existence of Inverses:**

• A function must be bijective (both injective and surjective) to have an inverse.
• Graphically, for a function to have an inverse, it must pass the Horizontal Line Test—no horizontal line intersects the graph more than once.

**Composition of Functions and Their Inverses:**

For a function $f$ and its inverse $f^{-1}$:

• $(f \circ f^{-1})(x) = x$
• $(f^{-1} \circ f)(x) = x$

**Example:**

Given $f(x) = 2e^{x}$ and its inverse $f^{-1}(x) = \ln\left(\frac{x}{2}\right)$, verify the compositions:

2. Analyzing the Product of Functions

The product of two functions introduces complexity in analysis, especially regarding domains, ranges, and differentiability. Understanding how functions interact when multiplied is essential for solving intricate problems.

**Domain Considerations:**

• The domain of $fg(x)$ is the intersection of the domains of $f(x)$ and $g(x)$.
• Points where either $f(x)$ or $g(x)$ is undefined are excluded from the domain of $fg(x)$.

**Range Analysis:**

Determining the range of $fg(x)$ often requires analyzing the behavior of both functions over their common domain. This can involve finding maximum and minimum values, asymptotes, and intercepts.

**Differentiability:**

If both $f(x)$ and $g(x)$ are differentiable, then $fg(x)$ is also differentiable. The derivative is given by the product rule:

**Example Problem:**

1. Let $f(x) = e^{x}$ and $g(x) = x^{2}$. Find the derivative of $fg(x)$.

**Solution:**

3. Power Functions and Their Properties

Power functions extend the concept of polynomials and exponential functions by allowing the output of a function to be raised to a power. Analyzing power functions involves understanding their growth rates, curvature, and behavior under various operations.

**Behavior Based on Exponents:**

• Even Exponents: The graph is symmetric about the y-axis.
• Odd Exponents: The graph is symmetric about the origin.
• Positive Exponents: The function increases as $x$ moves away from zero.
• Negative Exponents: The function decreases as $x$ moves away from zero.

**Growth Rates:**

• Higher exponents result in faster growth rates.
• Comparing $f(x) = x^{2}$ and $g(x) = x^{3}$, $g(x)$ grows faster for large values of $x$.

**Transformations:**

• Vertical Shifts: Adding or subtracting a constant shifts the graph up or down.
• Horizontal Shifts: Adding or subtracting a constant inside the function shifts the graph left or right.
• Scaling: Multiplying by a constant stretches or compresses the graph vertically.

**Example Problem:**

1. Determine the end behavior of $f(x) = x^{4}$ as $x \to \infty$ and $x \to -\infty$.

**Solution:**

• As $x \to \infty$, $f(x) = x^{4} \to \infty$.
• As $x \to -\infty$, $f(x) = x^{4} \to \infty$ (since the exponent is even).

4. Interdisciplinary Connections

The concepts of function notation and operations on functions extend beyond pure mathematics, finding applications in various disciplines such as physics, economics, biology, and engineering. Understanding these connections enhances the ability to apply mathematical principles to real-world problems.

**Physics:** Exponential functions model radioactive decay and population dynamics, while logarithmic functions describe phenomena like sound intensity and pH levels.

**Economics:** Functions representing cost, revenue, and profit often involve product and power functions to model complex relationships between variables.

**Biology:** Growth models for populations and the spread of diseases utilize exponential and logistic functions.

**Engineering:** Power functions are essential in designing structures, analyzing stress and strain, and optimizing performance parameters.

**Example Application:**

In electrical engineering, the relationship between voltage, current, and resistance is governed by Ohm's Law, which can be expressed using function notation:

Here, $V(x)$ is the voltage as a function of current $I(x)$, and $R$ is the constant resistance.

Understanding how to manipulate and analyze such functions is crucial for designing and troubleshooting electrical circuits.

5. Complex Problem-Solving Using Function Notation

Solving complex problems often requires the integration of multiple function operations, such as composing inverse functions, multiplying functions, and raising functions to powers. Mastery of these operations facilitates tackling multi-step problems encountered in higher-level mathematics and applied fields.

**Example Problem:**

1. Given $f(x) = 3e^{x}$ and $g(x) = \log_{10}(x)$, find $f^{-1}(gh(f(x)))$.

**Solution:**

• First, find $f(x)$: $f(x) = 3e^{x}$.
• Find $gh(y) = g(h(y))$. Assuming $h(y)$ is a function, but since $h$ is not defined, we'll assume $gh(x) = g(x) \cdot h(x)$. However, since the problem is unclear, we'll reinterpret it as the composition $g(h(f(x)))$ where $h$ is intended to be $f^{-1}$.
• Thus, $gh(f(x)) = g(f^{-1}(f(x))) = g(x)$.
• Now, $f^{-1}(gh(f(x))) = f^{-1}(g(x))$.
• Since $f^{-1}(x) = \ln\left(\frac{x}{3}\right)$, then:
$$ f^{-1}(g(x)) = \ln\left(\frac{g(x)}{3}\right) = \ln\left(\frac{\log_{10}(x)}{3}\right) $$

Comparison Table

Summary and Key Takeaways