One-to-One Functions and Inverses

Definition and Properties

One-to-one functions are fundamental in precalculus, especially when discussing invertibility. A function is one-to-one if each output value corresponds to exactly one input value.

• Invertible functions: A function f is invertible if there exists a function g such that f(g(x)) = x and g(f(x)) = x for all x in the domain.

• Inverse functions: f and g are inverses of each other if and only if f(g(x)) = x and g(f(x)) = x.

• Finding the inverse: To find the inverse of y = f(x), solve for x in terms of y, then interchange x and y.

Domain and Range:

• The domain of f^{-1} is the range of f.

• The range of f^{-1} is the domain of f.

Graphs: The graphs of y = f(x) and y = f^{-1}(x) are reflections of each other across the line y = x.

Properties of Exponents

Exponent Rules

Exponents are used to express repeated multiplication and are essential for understanding exponential and logarithmic functions.

• Product Rule:

• Quotient Rule:

• Power Rule:

• Power of a Product:

• Negative Exponent:

• Zero Exponent: (for )

• Fractional Exponent:

Logarithmic Functions

Definition and Properties

A logarithmic function is the inverse of an exponential function. The general form is y = \log_a x for x > 0, a > 0, and a \neq 1.

• Definition: means

• Exponential form:

• Domain:

• Range:

• Increasing: If

• Decreasing: If

Logarithm Properties

• Product Rule:

• Quotient Rule:

• Power Rule:

• Logarithm of 1:

• Logarithm of base:

• Change of Base:

Solving Exponential and Logarithmic Equations

Exponential Equations

To solve exponential equations, express both sides with the same base if possible, or use logarithms.

• Like bases: If , then

• General case: Take logarithms of both sides to solve for the variable.

Example:

• Solve . Since , .

• Solve . Take logarithms:

Logarithmic Equations

To solve logarithmic equations, use properties of logarithms to combine terms and isolate the variable.

Example:

• Solve . Combine:

Applications of Exponential and Logarithmic Functions

Compound Interest

Compound interest calculations use exponential functions to model the growth of investments.

• Compounded once per year:

• Compounded n times per year:

• Compounded continuously:

Example:

• If , , , compounded quarterly:

Exponential Growth and Decay

Exponential models describe processes that increase or decrease at rates proportional to their current value.

• Growth: , where

• Decay: , where

• Half-life:

• Doubling time:

Example:

• If , half-life is

Newton's Law of Cooling

Newton's Law of Cooling models the temperature change of an object in a surrounding medium.

• Where: is the surrounding temperature, is the initial temperature, is a constant.

pH and Logarithms in Chemistry

pH is a logarithmic measure of hydrogen ion concentration in a solution.

• For ,

Summary Table: Exponential and Logarithmic Properties

Additional info:

• Some examples and applications were inferred from standard precalculus curriculum and the context of the notes and questions.

• Definitions and formulas were expanded for completeness and clarity.