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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. Measurable Functions [09/13/2004] Can you provide some motivation for the definition of a measurable function? Meromorphic Functions [09/18/1998] What is a meromorphic function? The Meteorologists' Theorem [1/6/1995] Prove the "Meteorologists' Theorem": At any given moment, there are two diametrically opposite points on the (spherical?) Earth's surface where the temperatures are equal and the barometric pressure are equal. Multiplying Square Roots of Negative Numbers [11/25/2003] It seems to me there are two possible ways to interpret a problem like sqrt(-2) * sqrt(-2). One way I get 2 and the other way I get -2. Which solution is correct? What's wrong with the other one? Mystery Operation: What Do These Strange Symbols Mean? [03/09/2010] How do you evaluate the composition of two functions when you don't even recognize their operational symbols? Doctor Ian unravels the mystery behind weird symbols, explains their instructional purpose, and uses grouping parentheses to show how to compose functions according to the order of operations. Naming and Graphing Functions [06/16/1997] Can you tell me about the different families of functions? Natural Domain of a Function [03/12/1999] Some inputs don't make sense for some functions. Newton-Raphson Method [02/28/2000] How can I find the 1st, 2nd, 3rd iterations and the parameters of x^3 - 13.1x^2 + 48.48x - 46.62 using the Newton-Raphson method? Newton-Raphson Method [06/24/2009] Are there any equations that cannot be solved using the Newton-Raphson method, regardless of the initial estimate? Newton's Method for Finding Roots of Functions [06/03/2003] Is there an easy way to learn and understand this method or an example that would allow me to understand the steps taken to work through the problem? Nondifferentiable Functions [11/6/1994] In my calculus book, it mentions a function that is not differentiable at any point due to the fact that it is not smooth at any point. It does not go any farther, and I was interested in hearing more. Normalizing Ranges of Numbers [04/22/2002] How do I convert ranges like 5-200 to a scale of 1-10? O and o Functions [01/08/1999] How do big O notation and little o notation show the relation between two functions? Odd and Even Function Constructions [06/08/1998] If f is an odd function and g is an even function, how do we combine the two in various constructions (adding, multiplying, etc.)? Odd And Even Functions [01/11/1998] The differences between odd and even functions, and examples of each. One to One Correspondence Between Sets [04/04/1997] What is the one-to-one correspondence between the set of natural numbers and the set "S" = {1/3, 1/6, 1/12, 1/24, 1/48, ...}? One-to-one Function [02/05/1998] If A = (0,1) and B = (a,b) show that there is a one-to-one function from A onto B. ONTO and INTO [07/27/2001] What is the difference between ONTO and INTO when you describe a function? Onto (Surjective) Functions [11/03/1998] I don't understand onto functions. Would you please explain them in detail? Order of Transformations of a Function [11/30/2005] In what order would I perform transformations such as -f(x), f(-x), af(x), f(ax), f(x)+a, and f(x+a) if two or more are applied to f(x)? If I had f(ax+b), would I do the translation or the stretch first? Order of Transformations of a Function, Redux [01/31/2017] A teen gets confused when the graph of a transformation appears to disproportionately shrink its horizontal shift. Re-visiting his example, Doctor Peterson points out how the transformation shrunk even the shift. Order of Transformations of a Function, Yet Again [03/28/2017] Another teen notices curious translations at play when adding terms to absolute value functions. Doctor Peterson peels back the steps in order to reveal what's really at play — and how the effect holds for all combinations of horizontal transformations. Parameter vs. Constant [12/08/2002] Can you explain the difference between parameter and constant? Parameter vs. Temperature Formula [10/28/2002] I am trying to figure out a formula for frequency vs. temperature. The Phi Function [11/21/1998] What are the conditions on n,m so that phi(n*m) = phi(n)*phi(m)? What is phi(p^n*q^m)? Piecewise Function [10/23/2001] How do you graph the piecewise function? Piecewise Functions on TI Graphing Calculators [02/08/2005] I was wondering if you could tell me how to graph a piecewise function on a TI-83 plus graphing calculator? Polar Coordinates and Logarithmic Spirals [04/25/2004] The polar coordinate graph of a logarithmic spiral suggests that for a given angle, there is more than one possible r-value, since the graph continues to spiral around itself. How is it possible for an angle to be associated with more than one r? Possible Values of f(0) [04/19/2002] If f(x+y) = f(x)f(y), what is (are) the possible value(s) of f(0)? Product Notation [10/26/2000] Can the pi symbol also be used to mean multiplying all the terms in a series together? A Project on Cycloids [01/16/1999] Can you explain cycloids? How do you work with the parametric equations? What are their properties? How are they related to time? Proving One Function is Greater Than Another [11/1/1996] Prove that if n is a positive integer and x > 0, then x^n + 1/(x^n) > x^(n-1) + 1/[x^(n-1)]. Proving Subsets [07/01/2003] Given the function f:X --> Y and subsets A of X and B of Y, prove the following statements: A is a subset of f^{-1}(f(A)); f(f^{-1}(B)) is a subset of B. PSAT Math Question on Functions [10/17/2003] For all values of r, let *r be defined as *r = (r + 2)/2. If *4 = x, then *x = ? Purpose of Absolute Value and Piecewise Functions [12/09/2001] What is the purpose of absolute value functions and of piecewise functions? Quarter Circles [12/20/1996] Find a function that graphs a quarter of a circle in quadrants II and IV. Raised Cosines [01/28/2001] What are raised cosines, and how do they work? Range [04/02/2002] If f(x) = x-3/x+3, I know the domain to be all Reals except for -3. The range, however, is all Reals except for 1. I don't know how they get the 1 algebraically. Range and Codomain of a Relation [03/27/2004] What is the difference between the range and the codomian of a relation? Range and Domain of a Graph [02/12/1999] Determine the domain and range of the graph of "f" that starts at (- 2,3), goes down to (0,0), and ends at (3,4). Page: []

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