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 result 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 $f(x)$.

• Domain: The domain of a composite function $f(g(x))$ is all $x$ in the domain of $g$ such that $g(x)$ is in the domain of $f$.

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

Domain and Range of Relations

The domain of a relation is the set of all possible input values, while the range is the set of all possible output values. A relation is one-to-one if each input corresponds to a unique output.

• Domain: List all $x$-values in the set.

• Range: List all $y$-values in the set.

• One-to-One Test: A relation is one-to-one if no $y$-value is repeated for different $x$-values.

• Example: For $\{(-1,4), (1,4), (4,-7), (7,-8)\}$, domain is $\{-1,1,4,7\}$, range is $\{4,-7,-8\}$.

Inverse Functions

Finding and Graphing Inverses

An inverse function reverses the effect of the original function. The graph of an inverse is a reflection across the line $y=x$.

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

• Domain and Range: The domain of $f$ becomes the range of $f^{-1}$ and vice versa.

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

Transformations of Functions

Graphing with Transformations

Transformations shift, stretch, compress, or reflect the graph of a function. Key points and asymptotes help describe the graph.

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

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

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

• Example: $f(x) = (1/x)^2 - 2$ is a transformation of the reciprocal squared function.

Exponential and Logarithmic Functions

Properties and Simplification

Exponential and logarithmic functions are used to model growth, decay, and many real-world phenomena. Understanding their properties is essential for solving equations and simplifying expressions.

• Exponential Laws: $a^{m} \cdot a^{n} = a^{m+n}$, $(a^{m})^{n} = a^{mn}$

• Logarithm Laws: $\log_b(xy) = \log_b x + \log_b y$, $\log_b(\frac{x}{y}) = \log_b x - \log_b y$, $\log_b(x^k) = k \log_b x$

• Example: $\log_3(4x^2y^5) = \log_3 4 + 2\log_3 x + 5\log_3 y$

Solving Exponential and Logarithmic Equations

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

• Exponential Equations: Set bases equal if possible, or take logarithms of both sides.

• Logarithmic Equations: Combine logs using properties, then exponentiate both sides.

• Example: Solve $2^{3x} = 32$ by writing $32$as $2^5$, so $3x = 5$ and $x = \frac{5}{3}$.

Applications of Exponential and Logarithmic Functions

Compound Interest

Compound interest is calculated using exponential functions. The formula for compound interest is:

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

• Variables: $A$ = final amount, $P$ = principal, $r$ = annual interest rate, $n$ = number of compounding periods per year, $t$ = number of years.

• Example: If $P = 750$, $r = 0.049$, $n = 1$, $t = 5$, then $A = 750(1 + 0.049)^5$.

Exponential Decay and Half-Life

Radioactive decay and half-life problems use exponential decay formulas.

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

• 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	Equation	Example
Product Rule	$\log_b(xy) = \log_b x + \log_b y$	$\log_2(8 \cdot 4) = \log_2 8 + \log_2 4$
Quotient Rule	$\log_b(\frac{x}{y}) = \log_b x - \log_b y$	$\log_5(25/5) = \log_5 25 - \log_5 5$
Power Rule	$\log_b(x^k) = k \log_b x$	$\log_3(x^2) = 2 \log_3 x$

Key Concepts and Examples

• Composite Functions: $f(g(x))$ and $g(f(x))$

• Inverse Functions: $f^{-1}(x)$

• Transformations: Shifts, stretches, reflections

• Exponential Equations: $a^{x} = b$

• Logarithmic Equations: $\log_b x = y$

• Compound Interest: $A = P(1 + r/n)^{nt}$

• Half-Life: $A = A_0 (1/2)^{t/h}$

Additional info: These notes cover topics from Ch. 1 (Functions and Graphs), Ch. 3 (Exponential and Logarithmic Functions), and related applications, as indicated by the exam review questions.