Precalculus Exam 4 Review: Functions, Logarithms, Exponentials, and Applications

Functions and Their Properties

Function Composition and Evaluation

Function composition involves applying one function to the results of another. Evaluating functions at specific points and composing functions are foundational skills in precalculus.

• Function Evaluation: To evaluate $f(a)$, substitute $a$ into the function $f(x)$.

• Function Composition: The composition $f(g(x))$ means to substitute $g(x)$ into $f(x)$ wherever $x$ appears.

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

Domain and Range

The domain of a function is the set of all possible input values (x-values) for which the function is defined. The range is the set of all possible output values (y-values).

• Finding the Domain: Exclude values that make denominators zero or arguments of even roots negative.

• Finding the Range: Analyze the behavior of the function or use its graph.

• Example: For $f(x) = \frac{2x-1}{x}$, the domain is all real numbers except $x=0$.

One-to-One Functions

A function is one-to-one if each output is produced by exactly one input. This property is important for finding inverses.

• Horizontal Line Test: If any horizontal line crosses the graph more than once, the function is not one-to-one.

• Example: $f(x) = x^3$ is one-to-one, but $f(x) = x^2$ is not.

Inverse Functions

Finding and Graphing Inverses

The inverse of a function $f(x)$, denoted $f^{-1}(x)$, reverses the roles of inputs and outputs. Not all functions have inverses; only one-to-one functions do.

• Finding the Inverse: Solve $y = f(x)$ for $x$ in terms of $y$, then swap $x$ and $y$.

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

• Example: For $f(x) = \frac{2x+1}{x}$, solve $y = \frac{2x+1}{x}$ for $x$.

Transformations of Functions

Shifts, Reflections, and Stretches

Transformations change the position or shape of a graph. Common transformations include vertical and horizontal shifts, reflections, and stretches/compressions.

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

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

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

• Example: $f(x) = (1/x) - 2$ is a vertical shift of $f(x) = 1/x$ down by 2 units.

Exponential and Logarithmic Functions

Properties and Applications

Exponential and logarithmic functions are used to model growth, decay, and many real-world phenomena.

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

• Logarithmic Function: $f(x) = \log_a(x)$ is the inverse of $a^x$.

• Domain of Logarithms: The argument must be positive: $x > 0$.

• Example: $f(x) = \log(x^2 - 9)$ has domain $x^2 - 9 > 0 \implies x > 3$ or $x < -3$.

Logarithm Properties and Simplification

Logarithms have several key properties that allow for simplification and expansion.

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

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

• Power Rule: $\log_a(x^r) = r\log_a(x)$

• Example: $\log(x^2y) = 2\log(x) + \log(y)$

Solving Exponential and Logarithmic Equations

To solve equations involving exponentials or logarithms, use properties of exponents and logarithms to isolate the variable.

• Exponential Equations: $a^{f(x)} = a^{g(x)} \implies f(x) = g(x)$

• Logarithmic Equations: $\log_a(f(x)) = \log_a(g(x)) \implies f(x) = g(x)$

• Example: Solve $2^{3x} = 8$ by writing $8 = 2^3$, so $3x = 3 \implies x = 1$.

Applications: Finance and Exponential Decay

Loan Payment Formula

Loans are calculated using the present value formula for annuities:

• Formula: $P = \frac{M}{r} \left[1 - \left(1 + r\right)^{-n}\right]$ where $P$ is the principal, $M$ is the monthly payment, $r$ is the monthly interest rate, and $n$ is the number of payments.

• Application: Used to determine the maximum loan amount based on affordable payments.

Compound Interest

Compound interest grows an investment by applying interest to both the initial principal and accumulated interest.

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

• Example: If $P = 750$, $r = 0.05$, $n = 365$, $t = 3$, then $A = 750\left(1 + \frac{0.05}{365}\right)^{365 \times 3}$.

Exponential Decay and Half-Life

Radioactive decay and other processes are modeled by exponential decay functions.

• Formula: $A = A_0\left(\frac{1}{2}\right)^{t/h}$ where $A_0$ is the initial amount, $h$ is the half-life, and $t$ is the elapsed time.

• Example: If $A_0 = 11.3$ mg, $h = 16.6$ hours, $t = 10$ hours, then $A = 11.3\left(\frac{1}{2}\right)^{10/16.6}$.

Summary Table: Logarithm Properties

Property	Formula	Example
Product Rule	$\log_a(xy) = \log_a(x) + \log_a(y)$	$\log_2(8 \cdot 4) = \log_2(8) + \log_2(4)$
Quotient Rule	$\log_a\left(\frac{x}{y}\right) = \log_a(x) - \log_a(y)$	$\log_3\left(\frac{9}{3}\right) = \log_3(9) - \log_3(3)$
Power Rule	$\log_a(x^r) = r\log_a(x)$	$\log_5(25) = \log_5(5^2) = 2\log_5(5)$

Key Concepts for Exam Preparation

• Understand how to evaluate and compose functions.

• Be able to determine domain and range, and test for one-to-one functions.

• Know how to find and graph inverse functions.

• Apply transformations to basic functions and identify asymptotes.

• Master properties of logarithms and exponentials for simplification and solving equations.

• Apply formulas for finance (loans, compound interest) and exponential decay.