Math Notes Summary PDF
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This document is math notes summarizing topics such as solving inequalities, one-to-one functions, inverse functions, exponential equations, logarithmic functions, and exponential inequalities. It provides a brief overview of each topic and demonstrations of step-by-step solutions.
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\# Math Notes Summary \#\# Section 1: Solving Inequalities with Exponential Bases This section covers solving inequalities where the base is an exponential expression. The key is that if the base is \> 1, the inequality sign remains the same; if the base is between 0 and 1, the inequality sign rev...
\# Math Notes Summary \#\# Section 1: Solving Inequalities with Exponential Bases This section covers solving inequalities where the base is an exponential expression. The key is that if the base is \> 1, the inequality sign remains the same; if the base is between 0 and 1, the inequality sign reverses. Four examples are provided demonstrating the step-by-step solution process. \#\# Section 2: One-to-One Functions A one-to-one function is defined as a set of ordered pairs where each element in the domain is paired with only one element in the range, and vice-versa. Examples using mapping diagrams illustrate one-to-one and not one-to-one functions. The horizontal line test is introduced as a graphical method for determining if a function is one-to-one. \#\# Section 3: Inverse Functions This section defines inverse functions and provides steps to find them: 1\. Replace f(x) with y. 2\. Swap x and y. 3\. Solve for y. 4\. Replace y with f⁻¹(x). Several examples demonstrate this process for different function types. Some examples show cases where the inverse function does not exist because the original function fails the horizontal line test. \#\# Section 4: Exponential Equations This section defines exponential equations and explains the one-to-one property of exponential functions. It details the law of exponents and provides examples of solving exponential equations by making the bases the same and equating the exponents. \#\# Section 5: Logarithmic Functions Logarithmic functions are introduced as the inverse of exponential functions. The properties of logarithmic functions are described, including domain, range, vertical asymptote, and intercepts. An example shows plotting a logarithmic function and determining its domain and range, as well as finding intercepts and zeros. \#\# Section 6: Exponential Inequalities This section defines exponential inequalities and explains how to solve them based on the base of the exponential expression. The rules for solving depend on whether the base is \> 1 or between 0 and 1, similar to Section 1.
