Precalculus Study Notes: Functions, Domains, Exponential and Logarithmic Equations

Functions and Their Domains

Definition of a Function

A function is a relation that assigns each element in the domain to exactly one element in the range. The domain of a function is the set of all possible input values (usually x-values) for which the function is defined.

• Difference Quotient: The difference quotient is used to measure the average rate of change of a function and is foundational for calculus. It is defined as: $\frac{f(x+h) - f(x)}{h}$

• Finding the Domain: To find the domain of a function, identify all x-values for which the function produces a real output. Exclude values that cause division by zero or negative values under an even root.

Examples:

• For $f(x) = \frac{x-1}{x^2-16}$, exclude $x$ values that make the denominator zero: $x^2-16=0 \implies x=4$ or $x=-4$. So, domain is all real numbers except $x=4$ and $x=-4$.

• For $f(x) = \frac{2x+2}{x^2-7x+10}$, factor denominator: $x^2-7x+10=(x-5)(x-2)$. Exclude $x=5$ and $x=2$.

• For $f(x) = \sqrt{1-2x}$, require $1-2x \geq 0 \implies x \leq \frac{1}{2}$.

Exponential Functions

Definition and Properties

An exponential function has the form $f(x) = a^x$, where $a > 0$ and $a \neq 1$. Exponential functions model growth and decay in many real-world contexts.

• Solving Exponential Equations: To solve equations like $4^x = 32$, express both sides with the same base if possible: $4^x = (2^2)^x = 2^{2x}$ $32 = 2^5$ Set exponents equal: $2x = 5 \implies x = 2.5$

• For $5^x = 625$, since $625 = 5^4$, $x = 4$.

Logarithmic Functions

Definition and Properties

A logarithm is the inverse of an exponential function. The logarithm base $a$ of $x$ is written $\log_a x$ and answers the question: "To what power must $a$ be raised to get $x$?"

• Converting Between Exponential and Logarithmic Forms: $a^x = b \iff \log_a b = x$

• Examples:

  • $\log_t e$

  • $\log \pi = c$

  • $\ln N = e$

• Solving Logarithmic Equations: To solve for $x$ in $16^x = 7$, take logarithms of both sides: $x = \frac{\log 7}{\log 16}$

Summary Table: Domain Restrictions

Function	Domain Restriction	Reason
$\frac{x-1}{x^2-16}$	$x \neq 4, -4$	Denominator cannot be zero
$\frac{2x+2}{x^2-7x+10}$	$x \neq 2, 5$	Denominator cannot be zero
$\sqrt{1-2x}$	$x \leq \frac{1}{2}$	Radicand must be non-negative

Additional info:

• "BABY function" likely refers to basic exponential functions such as $f(x) = a^x$.

• "Possibilities" sections on the board refer to possible forms or solution strategies for exponential and logarithmic equations.