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- https://math.libretexts.org/Courses/Coastline_College/Math_C180%3A_Calculus_I_(Everett)/01%3A_Functions_and_Graphs/1.02%3A_Review_of_FunctionsIn this section, we provide a formal definition of a function and examine several ways in which functions are represented—namely, through tables, formulas, and graphs. We study formal notation and ter...In this section, we provide a formal definition of a function and examine several ways in which functions are represented—namely, through tables, formulas, and graphs. We study formal notation and terms related to functions. We also define composition of functions and symmetry properties. Most of this material will be a review for you, but it serves as a handy reference to remind you of some of the algebraic techniques useful for working with functions.
- https://math.libretexts.org/Courses/Reedley_College/Calculus_I_(Casteel)/01%3A_Functions_and_Graphs/1.01%3A_Review_of_FunctionsIn this section, we provide a formal definition of a function and examine several ways in which functions are represented—namely, through tables, formulas, and graphs. We study formal notation and ter...In this section, we provide a formal definition of a function and examine several ways in which functions are represented—namely, through tables, formulas, and graphs. We study formal notation and terms related to functions. We also define composition of functions and symmetry properties. Most of this material will be a review for you, but it serves as a handy reference to remind you of some of the algebraic techniques useful for working with functions.
- https://math.libretexts.org/Courses/Monroe_Community_College/MTH_210_Calculus_I_(Seeburger)/01%3A_Functions_and_Graphs/1.01%3A_Review_of_FunctionsIn this section, we provide a formal definition of a function and examine several ways in which functions are represented—namely, through tables, formulas, and graphs. We study formal notation and ter...In this section, we provide a formal definition of a function and examine several ways in which functions are represented—namely, through tables, formulas, and graphs. We study formal notation and terms related to functions. We also define composition of functions and symmetry properties. Most of this material will be a review for you, but it serves as a handy reference to remind you of some of the algebraic techniques useful for working with functions.
- https://math.libretexts.org/Courses/Cosumnes_River_College/Math_372%3A_College_Algebra_for_Calculus/11%3A_Appendix_-_Prerequisite_Function_Material/11.05%3A_Graphing_Functions_by_Point-PlottingIt’s time to look at functions graphically again, only this time we’ll do so with the notation defined in Section 1.4.
- https://math.libretexts.org/Courses/Laney_College/Math_3A%3A_Calculus_1_(Fall_2022)/01%3A_Functions_and_Graphs/1.02%3A_Review_of_FunctionsIn this section, we provide a formal definition of a function and examine several ways in which functions are represented—namely, through tables, formulas, and graphs. We study formal notation and ter...In this section, we provide a formal definition of a function and examine several ways in which functions are represented—namely, through tables, formulas, and graphs. We study formal notation and terms related to functions. We also define composition of functions and symmetry properties. Most of this material will be a review for you, but it serves as a handy reference to remind you of some of the algebraic techniques useful for working with functions.
- https://math.libretexts.org/Workbench/Book-_Precalculus_I_for_Highline_College_w/Rational_Inequalities_and_Equations_of_Circles/1.01%3A_Functions/1.1.06%3A_Transformation_of_FunctionsOften when given a problem, we try to model the scenario using mathematics in the form of words, tables, graphs, and equations. One method we can employ is to adapt the basic graphs of the toolkit fun...Often when given a problem, we try to model the scenario using mathematics in the form of words, tables, graphs, and equations. One method we can employ is to adapt the basic graphs of the toolkit functions to build new models for a given scenario. There are systematic ways to alter functions to construct appropriate models for the problems we are trying to solve.
- https://math.libretexts.org/Bookshelves/Precalculus/Book%3A_Trigonometry_(Sundstrom_and_Schlicker)/02%3A_Graphs_of_the_Trigonometric_Functions/2.01%3A_Graphs_of_the_Cosine_and_Sine_FunctionsThe most basic form of drawing the graph of a function is to plot points. One thing we can observe from the graphs of the sine and cosine functions is that the graph seems to have a “wave” form and t...The most basic form of drawing the graph of a function is to plot points. One thing we can observe from the graphs of the sine and cosine functions is that the graph seems to have a “wave” form and that this “wave” repeats as we move along the horizontal axis.
- https://math.libretexts.org/Bookshelves/Differential_Equations/A_Second_Course_in_Ordinary_Differential_Equations%3A_Dynamical_Systems_and_Boundary_Value_Problems_(Herman)/05%3A_Fourier_Series/5.03%3A_Fourier_Series_Over_Other_Intervals\[a_{n}=\dfrac{1}{\pi} \int_{-\pi}^{\pi}|x| \cos n x d x=\dfrac{2}{\pi} \int_{0}^{\pi} x \cos n x d x \label{5.37} \] =& \dfrac{a_{0}}{2}+\sum_{n=1}^{\infty}\left[a_{n}\left(\cos \dfrac{2 n \pi t}{b-a...\[a_{n}=\dfrac{1}{\pi} \int_{-\pi}^{\pi}|x| \cos n x d x=\dfrac{2}{\pi} \int_{0}^{\pi} x \cos n x d x \label{5.37} \] =& \dfrac{a_{0}}{2}+\sum_{n=1}^{\infty}\left[a_{n}\left(\cos \dfrac{2 n \pi t}{b-a} \cos \dfrac{2 n \pi a}{b-a}+\sin \dfrac{2 n \pi t}{b-a} \sin \dfrac{2 n \pi a}{b-a}\right)\right.\\[4pt] &\left.+b_{n}\left(\sin \dfrac{2 n \pi t}{b-a} \cos \dfrac{2 n \pi a}{b-a}-\cos \dfrac{2 n \pi t}{b-a} \sin \dfrac{2 n \pi a}{b-a}\right)\right] \\[4pt]
- https://math.libretexts.org/Courses/Cosumnes_River_College/Math_370%3A_Precalculus/01%3A_Relations_and_Functions/1.06%3A_Graphs_of_FunctionsIt’s time to look at functions graphically again, only this time we’ll do so with the notation defined in Section 1.4.
- https://math.libretexts.org/Courses/College_of_Southern_Nevada/Calculus_(Hutchinson)/01%3A_Functions_and_Graphs_(Precalculus_Review)/1.01%3A_Review_of_FunctionsTherefore, the range must be a subset of \(\{y\,|\,y≥5\}.\) To show that every element in this set is in the range, we need to show that for a given \(y\) in that set, there is a real number \(x\) suc...Therefore, the range must be a subset of \(\{y\,|\,y≥5\}.\) To show that every element in this set is in the range, we need to show that for a given \(y\) in that set, there is a real number \(x\) such that \(f(x)=(x−4)^2+5=y\). Second, since the range of \(f\) is a subset of the domain of \(g\), the output \(f(x)\) is an element in the domain of \(g\), and therefore it is mapped to an output \(g(f(x))\) in the range of \(g\).
- https://math.libretexts.org/Workbench/Algebra_and_Trigonometry_2e_(OpenStax)/09%3A_Trigonometric_Identities_and_Equations/9.02%3A_Solving_Trigonometric_Equations_with_IdentitiesIn this section, we will begin an examination of the fundamental trigonometric identities, including how we can verify them and how we can use them to simplify trigonometric expressions.
