Inverse, Exponential, and Logarithmic Functions: College Algebra Study Notes

Inverse, Exponential, and Logarithmic Functions

4.1 Inverse Functions

Inverse functions are essential in algebra for reversing the effect of a function. This section covers one-to-one functions, the definition and properties of inverse functions, and methods for finding and verifying inverses.

• One-to-One Functions: A function is one-to-one if each output value corresponds to only one input value. Formally, if $f(a) = f(b)$ implies $a = b$, then $f$ is one-to-one.

• Horizontal Line Test: A function is one-to-one if every horizontal line intersects its graph at most once.

• Inverse Function Definition: If $f$ is a one-to-one function, its inverse $f^{-1}$ satisfies:

  • $f(f^{-1}(x)) = x$ for every $x$ in the domain of $f^{-1}$

  • $f^{-1}(f(x)) = x$ for every $x$ in the domain of $f$

• Finding Inverses: To find the inverse of $f(x)$:

  1. Replace $f(x)$ with $y$

  2. Interchange $x$ and $y$

  3. Solve for $y$

  4. Rewrite $y$ as $f^{-1}(x)$

• Graphing Inverses: The graph of $f^{-1}$ is a reflection of the graph of $f$ across the line $y = x$.

Example: If $f(x) = 2x + 3$, then $f^{-1}(x) = \frac{x-3}{2}$.

4.2 Exponential Functions

Exponential functions model rapid growth or decay and are fundamental in many real-world applications. This section introduces their properties, graphs, and equations.

• Definition: An exponential function has the form $f(x) = a^x$, where $a > 0$ and $a \neq 1$.

• Domain and Range:

  • Domain: $(-\infty, \infty)$

  • Range: $(0, \infty)$ for $a > 1$

• Graphing: The graph passes through $(0,1)$ and has a horizontal asymptote at $y=0$.

• Reflections and Translations: Transformations such as $f(x) = a^{-x}$ reflect the graph across the $y$-axis.

• Solving Exponential Equations: If $a^x = a^y$, then $x = y$.

Example: $f(x) = 2^x$ is increasing, $f(x) = (1/2)^x$ is decreasing.

Compound Interest

Compound interest is a key application of exponential functions in finance. It models the growth of investments over time.

• Formula: $A = P\left(1 + \frac{r}{n}\right)^{nt}$, where:

  • $A$ = final amount

  • $P$ = principal

  • $r$ = annual interest rate

  • $n$ = number of compounding periods per year

  • $t$ = number of years

• Continuous Compounding: $A = Pe^{rt}$

Example: If $P = 1000$, $r = 0.05$, $n = 4$, $t = 10$, then $A = 1000\left(1 + \frac{0.05}{4}\right)^{40}$.

4.3 Logarithmic Functions

Logarithmic functions are the inverses of exponential functions and are used to solve equations involving exponents.

• Definition: For $a > 0$, $a \neq 1$, $\log_a x$ is the exponent to which $a$ must be raised to get $x$.

• Properties:

  • Product: $\log_a(xy) = \log_a x + \log_a y$

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

  • Power: $\log_a(x^r) = r\log_a x$

• Change-of-Base Formula: $\log_a x = \frac{\log_b x}{\log_b a}$

• Common Logarithm: $\log_{10} x$

• Natural Logarithm: $\ln x = \log_e x$

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

Solving Logarithmic Equations

Logarithmic equations can be solved by rewriting them in exponential form or using properties of logarithms.

• If $\log_a x = b$, then $x = a^b$.

• Use properties to combine or expand logarithmic expressions before solving.

Example: $\log_3(x-2) = 2$ implies $x-2 = 9$, so $x = 11$.

4.4 Evaluating Logarithms and the Change-of-Base Theorem

This section covers how to evaluate logarithms using calculators and the change-of-base theorem.

• Change-of-Base Theorem: $\log_a x = \frac{\log_b x}{\log_b a}$

• Use this formula to compute logarithms with any base using common or natural logarithms.

Example: $\log_2 20 = \frac{\log_{10} 20}{\log_{10} 2}$

4.5 Exponential and Logarithmic Equations

Equations involving exponentials and logarithms can be solved using their properties and by converting between forms.

• To solve $a^{f(x)} = a^{g(x)}$, set $f(x) = g(x)$.

• To solve $\log_a f(x) = \log_a g(x)$, set $f(x) = g(x)$.

• Use logarithms to solve for exponents: $a^x = b \implies x = \log_a b$.

Example: $2^x = 10 \implies x = \log_2 10$

4.6 Applications and Models of Exponential Growth and Decay

Exponential growth and decay models describe processes that increase or decrease at rates proportional to their current value.

• Exponential Growth: $y = y_0 e^{kt}$, where $k > 0$

• Exponential Decay: $y = y_0 e^{-kt}$, where $k > 0$

• Doubling Time: $t = \frac{\ln 2}{k}$

• Half-Life: $t = \frac{\ln 2}{k}$

Example: If a population doubles every 5 years, $k = \frac{\ln 2}{5}$.

Tables

Below is a summary table of logarithm properties:

Property	Equation	Description
Product	$\log_a(xy) = \log_a x + \log_a y$	Logarithm of a product equals the sum of logarithms.
Quotient	$\log_a\left(\frac{x}{y}\right) = \log_a x - \log_a y$	Logarithm of a quotient equals the difference of logarithms.
Power	$\log_a(x^r) = r\log_a x$	Logarithm of a power equals the exponent times the logarithm.

Additional info: These notes include worked examples, step-by-step solutions, and graphical illustrations to reinforce understanding of inverse, exponential, and logarithmic functions, as well as their applications in finance and science.