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 (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

• Rational Function: $f(x) = \frac{x-1}{x^2-16}$ Domain: All real numbers except where the denominator is zero. Set $x^2-16=0 \Rightarrow x=4$ or $x=-4$. So, domain is $x \neq 4, -4$.

• Rational Function: $f(x) = \frac{2x+2}{x^2-7x+10}$ Domain: Set denominator $x^2-7x+10=0 \Rightarrow (x-5)(x-2)=0$, so $x \neq 2, 5$.

• Radical Function: $f(x) = \sqrt{1-2x}$ Domain: Set $1-2x \geq 0 \Rightarrow x \leq \frac{1}{2}$.

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. $4^x = 32 \Rightarrow (2^2)^x = 2^5 \Rightarrow 2^{2x} = 2^5 \Rightarrow 2x = 5 \Rightarrow x = 2.5$

• Example: $5^x = 625 \Rightarrow 625 = 5^4 \Rightarrow x = 4$

Logarithmic Functions

Definition and Properties

A logarithm is the inverse of an exponential function. The logarithm $\log_b a$ answers the question: "To what power must b be raised to get a?"

• Basic Logarithmic Equations:

  • $\log_t e$

  • $\log \pi = c$

  • $\ln N = e$

• Converting Between Exponential and Logarithmic Forms:

  • Exponential: $b^x = a$

  • Logarithmic: $x = \log_b a$

• Solving Logarithmic Equations: Isolate the logarithm and convert to exponential form if needed.

Example

• Solve for x: $16^x = 7$ Take logarithms on both sides: $x \log 16 = \log 7 \Rightarrow x = \frac{\log 7}{\log 16}$

Summary Table: Domain Restrictions

Function Type	Restriction	Example
Rational	Denominator $\neq 0$	$f(x) = \frac{1}{x-3}$, $x \neq 3$
Radical (even root)	Radicand $\geq 0$	$f(x) = \sqrt{x-2}$, $x \geq 2$
Logarithmic	Argument $> 0$	$f(x) = \log(x-1)$, $x > 1$

Additional info:

• These notes cover topics from Precalculus including functions, domains, exponential and logarithmic equations, and difference quotients, which are foundational for calculus.

• Exam information (date and time) is included but not relevant to mathematical content.