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- https://math.libretexts.org/Courses/Cosumnes_River_College/Math_370%3A_Precalculus/05%3A_Further_Topics_in_Functions
- https://math.libretexts.org/Courses/Northeast_Wisconsin_Technical_College/College_Algebra_(NWTC)/05%3A_Exponential_and_Logarithmic_Functions/5.04%3A_Logarithmic_Equations_and_Inequalities\(f^{-1}(x) = \dfrac{e^{2x} - 1}{e^{2x} + 1} = \dfrac{e^{x} - e^{-x}}{e^{x} + e^{-x}}\). (To see why we rewrite this in this form, see Exercise 51 in Section 11.10.) The domain of \(f^{-1}\) is \((-\i...\(f^{-1}(x) = \dfrac{e^{2x} - 1}{e^{2x} + 1} = \dfrac{e^{x} - e^{-x}}{e^{x} + e^{-x}}\). (To see why we rewrite this in this form, see Exercise 51 in Section 11.10.) The domain of \(f^{-1}\) is \((-\infty, \infty)\) and its range is the same as the domain of \(f\), namely \((-1, 1)\).
- https://math.libretexts.org/Bookshelves/Precalculus/Precalculus_(Stitz-Zeager)/01%3A_Relations_and_Functions/1.02%3A_RelationsAll of Precalculus can be thought of as studying sets of points in the plane. With the Cartesian Plane now fresh in our memory we can discuss those sets in more detail
- https://math.libretexts.org/Bookshelves/Precalculus/Precalculus_(Stitz-Zeager)/08%3A_Systems_of_Equations_and_Matrices/8.04%3A_Matrix_ArithmeticWe previously showed how we can rewrite a system of linear equations as the matrix equation AX=B where A and B are known matrices and the solution matrix X of the equation corresponds to the s...We previously showed how we can rewrite a system of linear equations as the matrix equation AX=B where A and B are known matrices and the solution matrix X of the equation corresponds to the solution of the system. In this section, we develop the method for solving such an equation.
- https://math.libretexts.org/Courses/Lorain_County_Community_College/Book%3A_Precalculus_Jeffy_Edits_3.75/04%3A_Rational_Functions/4.02%3A_Graphs_of_Rational_FunctionsIn this section, we take a closer look at graphing rational functions. Previously, we learned that the graphs of rational functions may have holes in them and could have vertical, horizontal and slant...In this section, we take a closer look at graphing rational functions. Previously, we learned that the graphs of rational functions may have holes in them and could have vertical, horizontal and slant asymptotes.
- https://math.libretexts.org/Courses/Lorain_County_Community_College/Book%3A_Precalculus_Jeffy_Edits_3.75/03%3A_Polynomial_Functions/3.02%3A_The_Factor_Theorem_and_the_Remainder_TheoremSuppose we wish to find the zeros of an arbitrary polynomial. Even though we could use the 'Zero' command to find decimal approximations for these, we seek a method to find the remaining zeros exactly...Suppose we wish to find the zeros of an arbitrary polynomial. Even though we could use the 'Zero' command to find decimal approximations for these, we seek a method to find the remaining zeros exactly. The point of this section is to generalize the technique applied here. First up is a friendly reminder of what we can expect when we divide polynomials.
- https://math.libretexts.org/Courses/Lorain_County_Community_College/Book%3A_Precalculus_(Stitz-Zeager)_-_Jen_Test_Copy/11%3A_Applications_of_Trigonometry/11.09%3A_The_Dot_Product_and_ProjectionPreviously, we learned how add and subtract vectors and how to multiply vectors by scalars. In this section, we define a product of vectors.
- https://math.libretexts.org/Courses/Lorain_County_Community_College/Book%3A_Precalculus_Jeffy_Edits_3.75/01%3A_Relations_and_Functions/1.07%3A_TransformationsIn this section, we study how the graphs of functions change, or transform, when certain specialized modifications are made to their formulas. The transformations we will study fall into three broad c...In this section, we study how the graphs of functions change, or transform, when certain specialized modifications are made to their formulas. The transformations we will study fall into three broad categories: shifts, reflections and scalings, and we will present them in that order.
- https://math.libretexts.org/Courses/Lorain_County_Community_College/Book%3A_Precalculus_Jeffy_Edits_3.75/09%3A_Sequences_and_the_Binomial_Theorem/9.0E%3A_9.E%3A_Sequences_and_the_Binomial_Theorem_(Exercises)These are homework exercises to accompany Chapter 9 of Stitz and Zeager's "Pre-Calculus" Textmap.
- https://math.libretexts.org/Courses/Chabot_College/MTH_36%3A_Trigonometry_(Gonzalez)/01%3A_Foundations_of_Trigonometry/1.03%3A_The_Six_Circular_Functions_and_Fundamental_IdentitiesWe previously defined cos(θ) and sin(θ) for angles θ using the coordinate values of points on the Unit Circle. As such, these functions earn the moniker circular functions (we will start using the phr...We previously defined cos(θ) and sin(θ) for angles θ using the coordinate values of points on the Unit Circle. As such, these functions earn the moniker circular functions (we will start using the phrase `trigonometric function' interchangeably with the term `circular function'). It turns out that cosine and sine are just two of the six commonly used circular functions which we define in this Module.
- https://math.libretexts.org/Bookshelves/Precalculus/Precalculus_(Stitz-Zeager)/09%3A_Sequences_and_the_Binomial_Theorem/9.04%3A_The_Binomial_TheoremSimply stated, the Binomial Theorem is a formula for the expansion of quantities for natural numbers.
