Inverse Functions

Definition and Properties

Inverse functions are fundamental in mathematics, allowing us to 'reverse' the effect of a function. A function f has an inverse if and only if it is one-to-one (injective), meaning each output is produced by exactly one input.

• Definition: Functions f and g are inverses if and for all x in their domains.

• Graphical Interpretation: The graph of an inverse function is a reflection of the original function across the line .

Example: If , then its inverse is .

Finding Inverses

• Replace with .

• Solve for in terms of .

• Interchange and ; solve for to get .

Example: Find the inverse of , .

• Let

• Solve for :

• Interchange and :

• Thus,

Example: Find the inverse of .

• Let

• Interchange and :

• Thus,

Properties of Exponents

Exponent Rules

Exponents follow several key properties that simplify expressions and solve equations.

• Product of Powers:

• Quotient of Powers:

• Power of a Power:

• Power of a Product:

• Power of a Quotient:

Rational Exponents

Rational exponents represent roots and powers in a unified way.

Example:

Exponential Functions

Definition and Types

An exponential function has the form , where and .

• If , the function models exponential growth (increasing function).

• If , the function models exponential decay (decreasing function).

Graphs of Exponential Functions

• The domain is .

• The range is .

• The horizontal asymptote is .

Example Table: For :

Logarithmic Functions

Definition and Relationship to Exponentials

A logarithmic function is the inverse of an exponential function. For , :

• if and only if

Logarithmic Form:

Exponential Form:

Graphing Logarithms

• The domain is .

• The range is .

• The vertical asymptote is .

Example Table: For :

Properties of Logarithms

• Product Rule:

• Quotient Rule:

• Power Rule:

• Logarithm of 1:

• Logarithm of the Base:

• Change of Base Formula:

Applications of Exponential and Logarithmic Functions

Exponential Growth and Decay

Many real-world phenomena, such as population growth and radioactive decay, are modeled by exponential functions.

• General Exponential Model:

• If , the process is growth; if , it is decay.

Half-life: The time required for a quantity to reduce to half its initial value. For decay constant :

Compound Interest

• n times per year:

• Continuously compounded:

pH and Logarithms

The pH of a solution is a logarithmic measure of hydronium ion concentration:

Example: If pH = 7.68, then M.

Summary Table: Exponential and Logarithmic Properties

Additional info: These notes summarize key Precalculus concepts on inverses, exponents, exponentials, logarithms, and their applications, suitable for exam review.