College Algebra Study Guide: Functions, Exponential & Logarithmic Equations, and Systems

Functions and Their Properties

Definition of a Function

A function is a correspondence between two sets where each input from the domain is assigned exactly one output in the codomain. If $f$ is a function, $f(x)$ denotes the output for input $x$.

• Domain: The set of all possible input values ($x$).

• Range: The set of all possible output values ($f(x)$).

• One-to-One Function: A function $f$ is one-to-one if $f(a) = f(b)$ implies $a = b$.

Example: $f(x) = x^2$ is not one-to-one since $f(2) = f(-2) = 4$.

Finding the Inverse of a Function

The inverse function $f^{-1}(x)$ reverses the effect of $f(x)$. To find the inverse:

• Step 1: Replace $f(x)$ with $y$.

• Step 2: Interchange $x$ and $y$.

• Step 3: Solve for $y$ in terms of $x$.

• Step 4: Replace $y$ with $f^{-1}(x)$.

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

Composition of Functions

The composition of functions $f$ and $g$ is written as $(f ullet g)(x) = f(g(x))$.

• To find $(f ullet g)(x)$, substitute $g(x)$ into $f(x)$.

Example: If $f(x) = x^2$ and $g(x) = x + 1$, then $(f ullet g)(x) = f(g(x)) = (x + 1)^2$.

Exponential Functions

Definition and Properties

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

• Domain: All real numbers

• Range: All positive real numbers

• Graph passes through $(0, 1)$

• Growth or decay depends on $a$

Example: $f(x) = 2^x$ is an exponential growth function.

Graphing Exponential Functions

To graph $f(x) = a^x$:

• Make a table of values for $x$ and $f(x)$.

• Plot the points and sketch the curve.

x	$2^x$
-2	0.25
-1	0.5
0	1
1	2
2	4

Example: The graph of $f(x) = 2^x$ increases rapidly for positive $x$.

Logarithmic Functions

Definition and Properties

A logarithmic function is the inverse of an exponential function. It is defined as $y = \log_a x$ if and only if $a^y = x$, where $a > 0$, $a \neq 1$.

• Domain: $x > 0$

• Range: All real numbers

• Graph passes through $(1, 0)$

Example: $y = \log_2 x$ is the inverse of $y = 2^x$.

Properties of Logarithms

• Product Rule: $\log_a(MN) = \log_a M + \log_a N$

• Quotient Rule: $\log_a\left(\frac{M}{N}\right) = \log_a M - \log_a N$

• Power Rule: $\log_a(M^k) = k \log_a M$

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

Solving Logarithmic and Exponential Equations

To solve equations involving logarithms and exponents:

• Use properties of logarithms to combine or expand expressions.

• Convert between exponential and logarithmic forms.

• Apply algebraic techniques to isolate the variable.

Example: Solve $2^x = 8$:

$2^x = 8$ $2^x = 2^3$ $x = 3$

Population Growth and Exponential Models

Exponential Growth Formula

Population growth can be modeled by the formula:

$P(t) = P_0 e^{rt}$

• $P_0$ = initial population

• $r$ = growth rate

• $t$ = time

Example: If $P_0 = 1000$ and $r = 0.05$, then after $t = 10$ years:

$P(10) = 1000 e^{0.5} \approx 1648.72$

Systems of Equations

Solving Systems by Substitution and Elimination

A system of equations consists of two or more equations with the same variables. Solutions are values that satisfy all equations simultaneously.

• Substitution Method: Solve one equation for one variable and substitute into the other.

• Elimination Method: Add or subtract equations to eliminate a variable.

Example: Solve the system:

$x + y = 5$

$x - y = 1$

Add equations: $2x = 6 \Rightarrow x = 3$ Substitute: $3 + y = 5 \Rightarrow y = 2$ Solution: $(3, 2)$

Summary Table: Logarithm Properties

Property	Formula
Product Rule	$\log_a(MN) = \log_a M + \log_a N$
Quotient Rule	$\log_a\left(\frac{M}{N}\right) = \log_a M - \log_a N$
Power Rule	$\log_a(M^k) = k \log_a M$
Change of Base	$\log_a M = \frac{\log_b M}{\log_b a}$

Additional info:

• Some examples and explanations have been expanded for clarity and completeness.

• Tables have been recreated to summarize key points and provide sample values for graphing.