College Algebra: Functions, Exponents, and Logarithms – Exam Study Guide

Functions and Their Operations

Evaluating and Combining Functions

Functions are mathematical relationships that assign each input exactly one output. In College Algebra, you often work with function notation and operations such as addition, subtraction, multiplication, division, and composition.

• Function Notation: If $f(x)$ and $g(x)$ are functions, $f(g(x))$ denotes the composition of $f$ and $g$.

• Example: If $f(x) = -7x + 3$ and $g(x) = 2x + 5$, then $f(g(-2))$ means substitute $-2$ into $g$, then use that result as input for $f$.

• Domain: The set of all possible input values for which the function is defined. For example, for $f(x) = \frac{1}{2}(2x - 5)$, the domain is all real numbers, $(-\infty, \infty)$.

Additional info: When composing functions, always check the domain of the inner function and ensure the output is valid for the outer function.

Exponential and Logarithmic Functions

Properties and Applications

Exponential and logarithmic functions are essential in modeling growth, decay, and compound interest. Understanding their properties allows you to solve equations and interpret real-world scenarios.

• Exponential Function: $f(x) = a \cdot b^x$, where $a$ is the initial value and $b$ is the growth/decay factor.

• Compound Interest Formula: $A = P\left(1 + \frac{r}{n}\right)^{nt}$, where $A$ is the amount after $t$ years, $P$ is the principal, $r$ is the annual interest rate, and $n$ is the number of compounding periods per year.

• Logarithmic Function: $y = \log_b(x)$ is the inverse of $y = b^x$.

• Properties of Logarithms:

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

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

  • $\log_b(x^r) = r \log_b(x)$

• Example: Solve $125^{3x+4} = 25$ using logarithms:

  • Take log of both sides: $\log(125^{3x+4}) = \log(25)$

  • Apply power rule: $(3x+4)\log(125) = \log(25)$

  • Solve for $x$.

Additional info: Logarithms are used to solve for exponents in equations involving exponential growth or decay.

Inverse Functions

Finding and Interpreting Inverses

The inverse of a function reverses the roles of input and output. If $f(x)$ maps $x$ to $y$, then $f^{-1}(x)$ maps $y$ back to $x$.

• Definition: $f^{-1}(f(x)) = x$ for all $x$ in the domain of $f$.

• Example: For $f(x) = \sqrt{x + 7}$, solve $y = \sqrt{x + 7}$ for $x$ to find $f^{-1}(x)$:

  • Square both sides: $y^2 = x + 7$

  • Isolate $x$: $x = y^2 - 7$

  • So, $f^{-1}(x) = x^2 - 7$

Modeling with Functions

Cost Functions and Average Cost

Functions can model real-world scenarios such as business costs. The cost function $C(x)$ gives the total cost for producing $x$ items, while the average cost function $\overline{C}(x)$ gives the cost per item.

• Cost Function: $C(x) = \text{fixed cost} + (\text{variable cost}) \times x$

• Average Cost Function: $\overline{C}(x) = \frac{C(x)}{x}$

• Example: If the fixed cost is $40.00 and each bottle costs $6.50, then$C(x) = 40 + 6.5x$

• For $x = 150$, $C(150) = 40 + 6.5 \times 150 = 40 + 975 = 1015$; $\overline{C}(150) = \frac{1015}{150} \approx 6.77$

Transformations of Functions

Graphical Transformations

Transformations change the position or shape of a function's graph. Common transformations include shifts, reflections, and stretches.

• Horizontal Shift: $f(x - h)$ shifts the graph $h$ units right.

• Vertical Shift: $f(x) + k$ shifts the graph $k$ units up.

• Reflection: $-f(x)$ reflects the graph over the x-axis.

• Stretch/Compression: $a f(x)$ stretches the graph vertically by $a$.

• Example: If $g(x)$ is a transformation of $f(x) = |x|$, describe the changes by comparing the graphs.

Logarithmic Expressions and Equations

Expanding and Rewriting Logarithms

Logarithmic expressions can be expanded or rewritten using properties of logarithms. This is useful for simplifying expressions and solving equations.

• Sum and Difference: $\ln(e^x + 3)$ can be expanded using logarithm properties.

• Rewriting Forms: $p^w = Q$ can be rewritten as $w = \log_p Q$.

• Example: $\log_2 x = 17$ is equivalent to $x = 2^{17}$.

Exponential Growth Models

Population Growth

Exponential models describe situations where quantities grow by a constant percentage rate over time.

• Growth Model: $A(t) = A_0 e^{rt}$, where $A_0$ is the initial amount, $r$ is the growth rate, and $t$ is time.

• Example: If a city's population is 48,900 in 2018 and grows at 2.7% per year, $A(t) = 48,900 e^{0.027t}$.

• To predict the population in 2026 ($t = 2026 - 2018 = 8$): $A(8) = 48,900 e^{0.027 \times 8}$

Table: Logarithm Properties and Examples

Property	Formula	Example
Product Rule	$\log_b(xy) = \log_b(x) + \log_b(y)$	$\log_{10}(100 \times 10) = \log_{10}(100) + \log_{10}(10) = 2 + 1 = 3$
Quotient Rule	$\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)$	$\log_{10}\left(\frac{1000}{10}\right) = \log_{10}(1000) - \log_{10}(10) = 3 - 1 = 2$
Power Rule	$\log_b(x^r) = r \log_b(x)$	$\log_{2}(8^3) = 3 \log_{2}(8) = 3 \times 3 = 9$