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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. Equation for a W-Shaped Graph [08/01/1999] What sort of equation has a graph that is shaped like a 'W'? Equation Solving with Newton-Raphson Method [07/23/2004] Is there a way to solve F = A*(((1+i)^n) - 1)/i for i, such that we have i = ...? I'm thinking I need to use a logarithm, but I'm stuck. Equivalent Fractions, Exponential Implications [02/17/2011] A student wonders how x^(a/b) changes for different multiples of a and b, and what difference it makes if you express that function as the b-th root of x^a. Doctor Ali corroborates the student's suspicion that domains matter -- as does the order of operations. Evaluating Composite Functions [1/31/1996] If f(x) = 3, and g(x) = x^2 + 1 divided by the square root of (x^2 -1), how do you evaluate the composite function of f composed with g (f circle g)? Everyday Applications of Inverse Functions [11/06/2011] A student seeks real-world applications of inverse functions. Doctor Peterson starts with an overview before suggesting several examples. Expansion of (x+y)^(1/2) [06/07/1999] Is there a way to expand (x+y)^(1/2)? If so, how is it derived? Explaining Independent and Dependent Variables [10/25/2002] Can you elaborate on the definitions of independent and dependent variables? Explicit Inverses [04/21/1999] How do you solve x^x = 5 without graphing or guessing? Exponentially Varying Functions [11/11/2001] I'm imagining a function that doubles after a certain interval, then doubles again in half that time, doubles again in a quarter of the original interval, etc. Are there functions that behave this way? Express a Function in Terms of Another Function [09/18/1997] The altitude perpendicular to the hypotenuse of a right triangle is 12 cm. Express the length of the hypotenuse in terms of the perimeter. Fazed by Fourier Transform Phases [11/14/2011] Complex output from an inverse discrete Fourier transform (DFT) raises doubts: would discarding imaginary values mean losing important information? Doctor Douglas explains why the phase of each Fourier component matters. Find a Function, Integrals [12/17/1997] Suppose the graph of f has the formula f(x)=-x+1 for 0<=x<=1; x-1 for 1 0. Finding an Inverse Function [8/28/1996] What is the inverse function of f(x) = 1 - 2x? What is the general method for finding an inverse function? Finding the Arc Length of a Hanging Catenary [07/23/1997] A catenary is suspended between two equal poles 400 feet apart at equal height; it sags in the center 40 feet... Finding The Domain and Range of a Function [01/27/2001] How do I find the domain and range of a function? My problem is f(x) = x squared -2. Finding the Domain of a Function [09/16/1997] How do you find the domain of this function: f(x) = (4+x)/(x^2-9) ? Finding the Domain of a Function [02/10/2009] I'm having trouble finding the domain of a function. What is the domain and how does it work? Finding the Factorial of Non-Integers [01/06/2005] I thought the factorial function was only defined for positive integers, but I notice that my calulator returns a value for non-integers like 3.5! What's going on? Finding the Function [12/17/1996] Given five related functions which have only one variable each, find the function that contains same all five variables. Finding the Inverse of y = x^x [08/31/2006] I have been trying to find the inverse function for y = x^x and once I exchange x and y I can't find a way to solve for y. Can you help? Finding the Roots of a Function [11/4/1996] How do you find the positive roots of x^9+3x^8-5x^3+4x+6=0? Show that x^n- 1=0 has exactly 2 roots if n is even, and only 1 real root if n is odd Finding x for e^x = cos(x) [06/25/1999] How can I solve e^x = cos(x) for x? Find the Function (fog)(x) [07/02/2003] Let f(x) = 2x-1 and g(x) = (x+5)/2. Fixed-Point Theorem [05/02/2002] An intuitive reason why the theorem is true, and a proof. Floor and Mod Functions Discontinuous [10/10/2001] What is f(x) = x mod b for some fixed b? Formulas for N-Dimensional Spheres [10/26/2000] What is the next term in the sequence pi*r^2, (4/3)pi*r^3, ...? Would it be the formula for the volume of a 4-dimensional sphere? Fourier Transforms [10/19/2001] I know what the sinc function is, but I can't convert my expression (in terms of sin) to an expression (in terms of sinc). Fractional Bases in Exponential and Logarithmic Functions [04/06/2009] In the equation log_a(x) = a^x, can the base be a fraction? Doctor Ali introduces a student to the Newton-Raphson Method and LambertW function. Functional Equation [02/18/2008] If 2f(x) + f(1-x) = x^2 for all x, then f(x) = ? Functional Transformations Explained and Undone [09/09/2013] A student wonders why horizontal transformations behave "opposite" to their original functions. Doctor Peterson explains with one example shifted by addition, and another scaled by multiplication. Function Crossing X-Axis [07/11/2003] Find the condition that must be satisfied by k in order so that the expression 2x^2 + 6x + 1 + k(x^2 + 2) may be positive for all real values of x. Function Deduction from Composition [11/19/2013] Given one function and the result of its composition with a second function, a student wonders how to recover the second function. Doctor Peterson shows how substitution solves such problems; Doctor Rick follows up with a method that relies on the inverse function. Function Exponentiation Convention [04/21/2013] A student seeks clarity on the meaning of f^2(x) and other expressions that juxtapose functions and exponents. Doctor Vogler empathizes, outlining the differing interpretations, before recommending contextual clues — and pointing up a larger lesson about ambiguity. The Function g(x) = gcd(x, 10) [01/14/1999] Graph the function g(x) = gcd(x,10), where x is a positive integer. What is the range? Assign a probability that g(x) = r for each r in the range. 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? Page: []

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