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 TOPICS This page:   functions    Search   Dr. Math See also the Internet Library:   functions HIGH SCHOOL About Math Analysis Algebra    basic algebra    equations/graphs/      translations    linear algebra    linear equations    polynomials Calculus Complex Numbers Calculators/    Computers Definitions Discrete Math    permutations/    combinations Exponents    Logarithms Fibonacci Sequence/   Golden Ratio Fractals Functions Geometry    Euclidean/plane      conic sections/        circles      constructions      coordinate plane      triangles/polygons    higher-dimensional      polyhedra    non-Euclidean    practical geometry    symmetry/tessellations History/Biography Interest Logic Negative Numbers Number Theory Physics/Chemistry Probability Projects Puzzles Sequences/Series Sets Square/Cube Roots Statistics Transcendental   Numbers Trigonometry Browse High School Functions Stars indicate particularly interesting answers or good places to begin browsing. Selected answers to common questions:     Composition of functions.     Domain and range.     Inverse of a function.     Piecewise functions. Function Its Own Inverse [01/02/1998] Show that f(x) = X/X-1 is its own inverse. Functions [11/12/1994] I'm a first year algebra student having difficulty learning about functions. Exactly how do they work? Functions and Equations [02/10/2001] When is an equation also a function? How do you recognize a function? Functions and Function Tables [05/11/1999] I don't understand functions and function tables. Can you help me? Functions and Inverses [05/29/2002] Find sets A, B, and C, and functions f:A->B and g:B->C, such that (g o f) is both injective and surjective, but f is not surjective and g is not injective. Functions: Domain, Range, and Piecewise [08/31/1998] What are piecewise functions? What are open and closed points? How do you figure out the domain and range of a function without graphing it on a calculator? Functions of a Complex Variable [10/05/2002] Show that w=f(z*), where z*=x-iy and f is a differentiable function, is not an analytic function of z. Functions of Imaginary Numbers [7/31/1996] Does (ln i) itself exist? Where does e^iA = cos A + i sin A come from? Functions O(k) and Omega(k) [01/20/2001] Please explain the difference between the functions O(k) and Omega(k). Functions Possible for a Symmetric, Asymptotic Graph [02/02/2013] A student seeks an even function that would represent a graph with a horizontal asymptote and symmetry around the y-axis. Doctor Peterson suggests considering functions of the form 1/f(x) — or graphing f(x) by finding 1/y for every x. Functions that are their own inverses [02/28/1998] Can a function ever be its own inverse? Functions: The Very Idea [04/08/2014] A teen struggles to grasp what constitutes a function, and to reconcile the uniqueness of two functions that differ in notationally or computationally trivial ways. Doctor Peterson offers perspectives both abstract and concrete. Functions Without a Second Derivative [6/28/1996] What are some examples of functions of a real variable whose derivatives don't have derivatives? The Gamma Function and Its Derivative [05/26/1998] Does n! = the integral from 0 to infinity of (x^n)(e^-x)dx hold true for all real numbers? If so, can we find the derivative of n!? General Strategy for Questions on Functions [06/10/2004] If the function f satisfies the equation f(x + y) = f(x) + f(y) for every pair of real numbers x and y, what is (are) the possible value(s) of f(0)? Graphing Absolute Values [07/03/2002] I have a function, f(x), that I can graph, but I don't understand how to graph |f(x)|. Graphing f(2x) and f(|x|) [09/03/2003] Given f(x), how do you graph f(2x) and f(|x|)? Graphing Limits [09/12/2001] I was hoping you could explain the concept of "limits" and how to read them from graphs. Graphing Piecewise Functions [09/09/2001] I do not understand how to graph piecewise functions. Graphing Polynomial Functions in Factored Form [07/07/2004] When graphing polynomial functions, why do factors of even multiplicity cause the graph to not cross the x-axis, while factors of odd multiplicity cause it to cross? Graphing Rational Functions and Vertical Asymptotes [03/28/2008] When working with rational functions such as y = (x-5)/(x-3), how do you know if the graph curves up or down at the vertical asymptote? Graphing Reciprocal Functions [11/01/2001] I know what a reciprocal is, but I don't know how to graph it. Graphing Sin and Cosine Functions [5/24/1996] I really need some help in finding the graphs of the following functions... Graphing the Absolute Value/Square Root of a Function [07/26/2002] Can you illustrate and discuss how taking the absolute value or the square root of a function affects the graph of the function? Graphing Two Functions With One Equation [10/21/2003] I have a question on graphing functions. I am just wondering if it is possible to use one equation to graph two functions? Graphing Two Functions with One Equation? [10/21/2003] Suppose I have two functions, like y = sqrt(x) - 4 and y = sqrt(x) - 8. Is there a single equation that would produce the same graph as the two functions graphed together? Graph of Circle as a Function [05/30/1998] The graph of a circle is not a function because it fails the vertical line test. How could you make it a function? Graphs of Inverse Functions [07/25/2002] Explain why, if y=f(x) and y=g(x) are inverses, the graph of either function is the graph of the other reflected across the line y=x. Graphs of Step Functions [11/29/2003] Why do some graphs move in steps rather than smooth lines? What causes that? Graph with f(x) [10/17/2001] Sketch the equations: y = f(x)+ 2; y = f(x-3); y = 2f(x). Greatest Integer Equation [08/06/2003] I am trying to correctly interpret [[x]]^2 + [[y]]^2 = 1, where f(x)= [[x]], is the Greatest Integer function. Greatest Integer Function [12/04/2002] How can I explain greatest integer function to my home-schooled son? Greatest Integer Functions [09/27/1998] Can you help me solve for the graph of [y]=[x], where [] is the greatest integer function? The Heaviside Step Function [09/03/2003] How can I construct a term that is zero when the input is below some limit, but nonzero above it? How Are Functions and Expressions Related? [07/09/2004] What is the relationship between a function and an expression? I don't see any relationship, they are two completely different things. How to Evaluate 4!!!! [10/16/2003] Our Calculus 2 teacher has challenged us to find the exact value for 4!!!!, but the number is too large to reasonably calculate by hand. How to Find the Range of a Function [02/25/1998] How do you find the range of a function like g(x) = (x+1)/(x^2-1)? How to Solve Equations with No Analytic Solution Method [10/26/2005] A discussion of solving equations that can't be solved analytically by using iterative estimation methods including bisection, false position, and Newton's method. The equation x(e^x) = 3 is used as an example. Hyperbolic Functions [06/27/1998] What does the inverse of sinh or cosh mean? Hyperbolic Functions [10/14/1998] Can you explain hyperbolic functions? How are they defined? How do you evaluate them? Page: []

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