College Algebra Exam 3 Review: Inverses, Exponential and Logarithmic Functions, and Applications

Inverse Functions and One-to-One Functions

Definition and Identification

An inverse function reverses the effect of the original function. A function is one-to-one if each output is produced by exactly one input.

• To find the inverse of a relation: Swap the x- and y-values in each ordered pair.

• To determine if a function is one-to-one: Use the horizontal line test (if any horizontal line crosses the graph more than once, the function is not one-to-one).

Example: For $f(x) = 5x - 4$, to find the inverse, solve for $x$ in terms of $y$:

$y = 5x - 4 \implies x = \frac{y + 4}{5}$, so $f^{-1}(x) = \frac{x + 4}{5}$

Graphs and Properties of Functions

Graphing and Analyzing Functions

Graphing helps visualize properties such as intercepts, symmetry, and one-to-one behavior.

• Vertical Line Test: Determines if a graph represents a function.

• Horizontal Line Test: Determines if a function is one-to-one.

Example: The graph of $f(x) = -x^2 + 2$ is a downward-opening parabola, which is not one-to-one because horizontal lines intersect it more than once.

Exponential and Logarithmic Functions

Definitions and Properties

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

• Logarithmic Function: $f(x) = \log_a x$, the inverse of the exponential function

• Properties:

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

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

  • $\log_a (x^r) = r \log_a x$

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

Solving Exponential and Logarithmic Equations

• To solve $a^x = b$, take the logarithm of both sides: $x = \log_a b$

• To solve $\log_a x = b$, rewrite as $x = a^b$

Example: Solve $3^x = 27$:

$x = \log_3 27 = 3$

Applications of Exponential and Logarithmic Functions

Compound Interest

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

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

Example: If $P = 5000$, $r = 0.12$, $n = 4$, $t = 2$, then $A = 5000\left(1 + \frac{0.12}{4}\right)^{4 \times 2}$

Population Growth and Decay

• Exponential Growth: $P(t) = P_0 e^{kt}$, where $k > 0$

• Exponential Decay: $P(t) = P_0 e^{-kt}$, where $k > 0$

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

Logarithmic Equations and Properties

Solving and Simplifying Logarithmic Expressions

• Combine logarithms using properties to simplify expressions.

• Express as a single logarithm if possible.

Example: $\log_2 8 + \log_2 4 = \log_2 (8 \times 4) = \log_2 32 = 5$

Change of Base Formula

• $\log_a b = \frac{\log_c b}{\log_c a}$ for any positive base $c \neq 1$

Example: $\log_2 10 = \frac{\log_{10} 10}{\log_{10} 2} = \frac{1}{0.3010} \approx 3.32$

Domain and Asymptotes of Logarithmic Functions

Finding Domain and Vertical Asymptotes

• The domain of $f(x) = \log_a (x - h)$ is $x > h$

• The vertical asymptote is at $x = h$

Example: For $f(x) = \log_2 (x + 4)$, domain is $x > -4$, vertical asymptote at $x = -4$

Summary Table: Logarithmic Properties

Property	Equation
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	$\log_a b = \frac{\log_c b}{\log_c a}$

Applications and Word Problems

Exponential Growth/Decay in Context

• Population models, compound interest, and radioactive decay are common applications.

• Set up equations based on the context and solve for the unknown variable.

Example: If $P = 2000e^{0.08t}$, to find when $P = 3000$:

$3000 = 2000e^{0.08t} \implies e^{0.08t} = 1.5 \implies 0.08t = \ln 1.5 \implies t = \frac{\ln 1.5}{0.08}$

Additional info: These notes cover topics from Ch. 7 (Inverse Functions), Ch. 9 (Exponential and Logarithmic Functions), and related applications, as outlined in the College Algebra curriculum.