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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. Implicit Functions [11/26/1997] Please give me a definition and several examples of an implicit function. Importance of Linear Functions [02/06/2002] What is the importance of linear functions in the real world, and how are they used? Infinity as a Skolem Function [10/28/2000] Is infinity an absolute concept, a relative concept, or both? Inner Product and L-2 Distance [11/14/2001] Why is "distance between two functions" calculated by multiplying f(x) and g(x) and then integrating with respect to x within the defined domain? Integer Iteration Function [12/24/2003] Let X be a positive integer, A be the number of even digits in that integer, B be the number of odd digits and C be the number of total digits. We create the new integer ABC and then we apply that process repeatedly. We will eventually get the number 123! How can we prove that? Integrating X^x, Closed Form [10/31/1996] How do you express the equation y = xcosx in terms of y? Interesting Continuity Question [09/01/2005] Is every function continuous at some point in its domain? Intermediate Value Theorem [09/19/2002] How can we prove by the intermediate value theorem that there is a point on the path that a hiker will cross at exactly the same time of the day hiking up and returning? Interpolation [03/06/2003] I cannot remember the formula for interpolating between two numbers. Inventing an Operation to Solve x^x = y [02/19/2000] Can we solve problems like x^x by inventing an operation higher than exponentiation? Can the "center" of an operation be defined? Inverse Functions: Arcsec(x) [02/11/1999] If y = arcsec(x) then what does x equal? How is this problem related to inverse functions? Inverse Functions in Real Life [01/25/2002] I would like to know if you have a good example of how inverse functions would be used in real life. Inverse Functions One-Way Only [06/01/1999] Are there two functions, f and g, such that (f of g)(x) = x but (g of f)(x) does not equal x? Inverse Functions - Which Statement is True? [09/20/1998] Does (f o g)^(-1) equal f^(-1) o g^(-1) or g^(-1) o f^(-1)? How can you prove it? Inverse of a Function vs. Inverse Proportionality [11/20/2002] Finding the inverse of a function and graphing yields a graph that has been reflected in the line y = x relative to the function. Inverse proportionality, however, yields a reciprocal relation graphically. Why do these two things have similar names yet mean different things? Inverse of a Multivariate Function [05/30/2002] Let f:NxN -> N such that f(x,y) = 2^x(2y + 1) - 1 for all natural numbers x, y. Let the inverse of f, g be given by g:N -> NxN. Find the inverse of the function g. Inverse of arg(z) [10/10/2003] What is the inverse of the function arg(z)? The Inverse of the Absolute Value Function [12/12/2008] What is the inverse of the absolute value function? I know the range must be restricted if the inverse is to be a function. Inverses [06/05/2001] What is an inverse? Inverses and Reciprocals of Functions [09/28/2005] What is the difference between f^(-1)(x) and f(x)^(-1)? Inverses of Functions, and Inverse Functions [02/12/2014] A student's textbooks give her conflicting definitions about functions and their inverses. Doctor Peterson makes sense of the disagreements by providing the bigger picture. Inverses of Trigonometric Functions [08/07/2002] How can the functions cosine and sine have inverses of arcsin and arccos respectively? Inverse Statements [08/20/1998] Say there are 3 + 4 times as many girls as boys. Is this the same as saying there are -3 + 1/4 as many boys as girls? Inverse Trigonometric Ratios [07/01/2003] Why aren't the inverse trigonometric ratios equal to 1/(the ratio), as is the case for numbers? Invertible Functions [06/06/2003] Consider as a function from R -> R (Real) and say whether the function is invertible: h(x) = (sgn x)* sqrt(abs(x)) where sgn is +1 if x is positive, -1 if x is negative, and 0 if x is 0. Inverting Functions [07/19/2002] To find the inverse of a function y=f(x), do I interchange the variables x and y, or do I solve for x in terms of y? Inverting, Subverted [05/03/2015] A teen wonders why interchanging a function's variables does not lead to its inverse. With several examples and two different approaches, Doctor Peterson disambiguates some commonly conflated notions. Irregular Sinusoidal Curves [08/24/2004] Can you create a sinusoidal function with a fixed period such that the time span between the max and min is not the same as the time span between the min and max? Is the Function Invertible? [09/22/1997] For the following functions f(x) decide if the function is invertible as a function from R to R... Is the Inverse a Function? [9/2/1996] How do I find the inverses and determine whether the inverses are functions? An Iterative Method of Calculating Pi [06/09/2004] I recently saw a method of calculating pi that involves an iterative function, P(n + 1) = P(n) + sin(P(n)) where P(n) is the approximation of pi at the nth iteration. Linear Equations and Standard Form [04/04/2002] What do the variables mean in the standard form of a linear equation (ax+by=c)? Linearity and Concavity [12/07/2003] Why does a linear function have no concavity? Line Crossing Horizontal and Vertical Asymptotes [03/16/1999] For the function y = (5x+1)/(x^2-1), why does the line between its two vertical asymptotes and one horizontal asymptote cross? Line of Best Fit, and Flight [02/04/2015] What do "extrapolation" and "interpolation" mean? What are their purpose? Doctor Peterson defines the terms, then gives real-world examples based on tracking the flight of his son's recent airplane trip. Lipschitz Continuous Functions [04/27/2008] Show that a Lipschitz continuous function is uniformly continuous on a subset and that the converse is not necessarily true. Also, give an example to show that Lipschitz continuous functions are not necessarily differentiable. Logs of Complex Numbers [02/11/2004] Give an example showing that Log(z1/z2) does not equal Log(z1) - Log(z2) where z1 and z2 are complex numbers. Manipulating Roots [08/08/1998] If a and b are the roots of the equation 3x^2 - 5x + 1 = 0, find the equation whose roots are a/b and b/a. Max and Min of Functions without Derivative [04/22/2003] Is there a general way to find the maximum and minimum of cubic functions without using derivatives? Maximizing Output of a Restricted Function [11/1/1996] Create a function whose domain is restricted to complex numbers but whose range is real, that is, non-constant, has no constant term, and contains no number greater than 3. Page: []

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