Study Notes on Exponential and Logarithmic Functions

Analysis of Exponential Decay and End Behavior

The example figure illustrates a portion of the graph of a function $f$ with the x-axis ( $y = 0$ ) as a horizontal asymptote.
The function exhibits exponential decay, meaning its outputs decrease as input values increase.
Evaluation of end behavior based on the provided answer choices: - Option A: $\lim_{x \to \infty} f(x) = -\infty$ - Analysis: As $x$ increases without bound (approaches positive infinity), the graph shows the function values approach the horizontal asymptote at $0$ . This option is incorrect. - Option B: $\lim_{x \to \infty} f(x) = \infty$ - Analysis: Similar to Option A, as the function moves to the right, the value approaches $0$ , not infinity. This option is incorrect. - Option C: $\lim_{x \to -\infty} f(x) = 0$ - Analysis: As $x$ decreases without bound (moves to the left), the function values are increasing without bound. The value of $0$ is only approached as $x \to \infty$ . This option is incorrect. - Option D: $\lim_{x \to -\infty} f(x) = \infty$ - Analysis: As we look to the left of the graph, the function is increasing without bound. This describes exponential decay approaching a horizontal asymptote on the right while growing toward the left. This is the correct description.

General Characteristics of Exponential Functions

An exponential function in its general form is written as: - $f(x) = a \times b^x$
Initial Value ( $a$ ): The value of the function when $x = 0$ . The value of $a$ cannot be zero because multiplying by zero would result in a constant output of zero.
Base or Common Ratio ( $b$ ): This represents the multiplier for each unit increase in $x$ . - Constraints: $b$ must be positive ( $b > 0$ ) and cannot equal one ( $b \neq 1$ ). - If $b = 1$ , the output remains constant regardless of the value of $x$ .
Exponential Growth vs. Exponential Decay: - Exponential Growth: Occurs when $a > 0$ and $b > 1$ . As you continue to multiply by a base greater than one, the output increases. - Exponential Decay: Occurs when $a > 0$ and $0 < b < 1$ . As you continue to multiply by a fractional value between zero and one, the output decreases and approaches a value of zero.
End Behavior Patterns for General Form: - For an equation in general form, as input values increase or decrease without bound, there are only three possible behaviors for output values: 1. Increase without bound ( $\infty$ ). 2. Decrease without bound ( $-\infty$ ). 3. Get arbitrarily close to zero ( $0$ ).

Constructing Exponential Equations

Exponential models can be generated using two primary methods: - Method 1: Initial Value and Ratio - From the graph in the first example, the point $(0, 6)$ is observed. This identifies the initial value ( $a$ ) as $6$ . - By looking at the next point, as we move one unit over ( $x = 1$ ), the output drops from $6$ to $3$ . This indicates the value was cut in half, making the ratio ( $b$ ) equal to $\frac{1}{2}$ . - The resulting equation is $y = 6 \times \left(\frac{1}{2}\right)^x$ . - Method 2: Solving a System of Equations - Use two known input-output pairs ( $(x_1, y_1)$ and $(x_2, y_2)$ ). - Example pairs: $(0, 6)$ and $(1, 3)$ . - Set up the equations: 1. $6 = a \times b^0$ 2. $3 = a \times b^1$ - Since any value (other than zero) raised to the power of zero is one ( $b^0 = 1$ ), the first equation simplifies to $a = 6$ . - Substitute $a$ into the second equation: $3 = 6 \times b^1 \implies 3 = 6b \implies b = \frac{3}{6} = \frac{1}{2}$ . - Both methods yield the consistent result: $y = 6 \times \left(\frac{1}{2}\right)^x$ .

Inverse Functions and Invertibility

Definition of Inverse Function: An inverse function is a "reverse mapping" of a function. Effectively, the input ( $x$ ) and output ( $y$ ) values switch places.
Graphical Representation: The graph of an inverse function is the reflection of the original function's graph over the identity line $y = x$ .
Composition Rule: The composition of a function and its inverse results in the identity function, meaning they "undo" or cancel each other out: $f(f^{-1}(x)) = x$ .
The Concept of Invertibility: - A function $f$ is invertible on a specified domain if every output value is mapped from a unique input value. - Example (Non-invertible): The function $f(x) = x^2$ . At $x = -2$ , the output is $4$ . At $x = 2$ , the output is also $4$ . When the relationship is reversed to create $x = y^2$ , an input of $4$ leads to two possible outputs ( $2$ and $-2$ ), which is not a function. - Restricting the Domain: A function can be made invertible by restricting its domain. For $f(x) = x^2$ , restricting the domain to $x \ge 0$ produces the invertible inverse function $f^{-1}(x) = \sqrt{x}$ . - Monotonicity Rule: If a function is strictly increasing or strictly decreasing on its entire domain, it is guaranteed to be invertible. Exponential functions (always increasing or always decreasing) are therefore always invertible on their domains.

Finding and Evaluating Inverses of Exponential Functions

Example Problem: Find the inverse of the function described by points $(1, 3)$ , $(2, 9)$ , $(3, 27)$ , which has a horizontal asymptote at the x-axis. - Step 1: Identify the Function. As $x$ increases by $1$ , the output is multiplied by a common ratio of $3$ . Thus, $b = 3$ . Using the point $(1, 3)$ : $3 = a \times 3^1 \implies a = 1$ . The function is $f(x) = 3^x$ . - Step 2: Swap Variables. To find the inverse, switch $x$ and $y$ : $x = 3^y$ . - Step 3: Isolate Output. To isolate $y$ from the exponent, use the inverse operation of an exponential base, which is a logarithm. Apply log base $3$ to both sides: - $\log_3(x) = \log_3(3^y)$ - $y = \log_3(x)$ - The inverse function is $f^{-1}(x) = \log_3(x)$ .
Evaluating Inverse Values (e.g., Evaluate $f^{-1}(3)$ ): - Conceptual Approach: Since inputs and outputs switch, finding $f^{-1}(3)$ is equivalent to finding what value of $x$ in the original function gives back an output of $3$ . Since $f(1) = 3$ , then $f^{-1}(3) = 1$ . - Direct Evaluation: Use the inverse function expression $f^{-1}(x) = \log_3(x)$ . Substituting $3$ for $x$ gives $\log_3(3)$ . Since $3^1 = 3$ , the value is $1$ .

Logarithmic Expressions Review

Logarithmic Definition: A logarithmic expression $\log_b(c)$ represents the value (exponent) that the base $b$ must be raised to in order to obtain the value $c$ .
Formal Relationship: $\log_b(c) = a \iff b^a = c$ .
Core Concept: A logarithm is, essentially, an exponent.
Common Logarithm: When the base of a logarithmic expression is not specified, it is understood to be a base of $10$ . Notation: $\log(x) = \log_{10}(x)$ . - Example: $\log(100) = 2$ , because $10^2 = 100$ .
Practice Evaluations: - Example 1: $\log_2(8)$ . Asking: $2^? = 8$ . Since $2^3 = 8$ , the value is $3$ . - Example 2: $\log_5(1)$ . Asking: $5^? = 1$ . Since $5^0 = 1$ , the value is $0$ . - Example 3: $\log_9(3)$ . Asking: $9^? = 3$ . Since $\sqrt{9} = 3$ , and a square root is an exponent of $\frac{1}{2}$ , the value is $\frac{1}{2}$ .

Logarithmic Properties

Logarithmic properties correspond to exponent properties and are designed to simplify mathematical calculations by reducing the complexity of operations.
Product Property: The logarithm of a product is equal to the sum of the logarithms of the factors. - $\log_b(x \times y) = \log_b(x) + \log_b(y)$
Quotient Property: The logarithm of a quotient is equal to the difference between the logarithm of the numerator and the logarithm of the denominator. - $\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)$
Power Property: The logarithm of an argument raised to an exponent is equal to the exponent multiplied by the logarithm of the argument. - $\log_b(x^n) = n \times \log_b(x)$ - Logic: $\log_b(x^3)$ is $\log_b(x \times x \times x)$ . Using the product property, this is $\log_b(x) + \log_b(x) + \log_b(x)$ , which equals $3 \times \log_b(x)$ .

The graph of an exponential decay function approaches a horizontal asymptote at the x-axis ( $y=0$ ) as the input values increase.
The correct end behavior for exponential decay can be described as $ext{lim}_{x o - ext{infinity}} f(x) = ext{infinity}$ , indicating output increases without bounds to the left of the graph.
An exponential function can be written in the form $f(x) = a imes b^x$ , where $a$ represents the initial value and $b$ is the base which must be positive and not equal to one.
Exponential growth occurs when $a > 0$ and $b > 1$ , while exponential decay occurs when $0 < b < 1$ .
The end behavior of exponential functions can be categorized into only three patterns: increase without bound, decrease without bound, or approach zero.