Functions

• All About Functions [Bonie, 11/06/1996]
  Could you please explain functions?

• Definition of the Signum Function [Little, 05/31/2000]
  Can you give me a simple definition of the signum function, and any practical examples of its usage?

• Exp, Log, and Ln Functions Explained [Ojanen, 7/31/1996]
  What is the exp function? When is it needed? Also, how do I calculate Log and Ln functions with basic arithmetic and logic?

• x Factorial and the Gamma Function [Basham, 05/29/1998]
  What is x! when x is 0, negative, or not a whole number?

• Function Machine [Johnson, 10/26/1996]
  How do you find the domain and range of a function?

• Function Tests [Smith, 02/19/1997]
  What is the reasoning behind the vertical and horizontal line tests?

• Interval Notation [Kelly, 4/1/1996]
  I need to learn about interval notation in terms of domain and ranges.

• Mapping Functions in the Real World [Elizee, 3/20/1995]
  What is the purpose of learning to map a function? What is it used for in the real world?

• Rational Inequality [Alan, 10/09/2001]
  Solve this rational inequality and give an answer in interval notation: -5/(3h+2) greater than or equal to 5/h.

• Sometimes, Always, or Never True? [Jon, 02/12/2002]
  Is this statement always, sometimes, or never true: f(g(x))=g(f(x)) ?

• What is a Function? [Anjali, 06/14/2001]
  I've read many definitions and I've asked many teachers, but I still don't completely understand.

• Why is Zero the Limit? [Katie, 02/25/2002]
  Why is zero called the limit of the terms in the sequence the limit of 1 over n, as n approaches infinity, equals zero?

• Nondifferentiable Functions [Finin, 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.

• Functions [Mark M., 11/12/1994]
  I'm a first year algebra student having difficulty learning about functions. Exactly how do they work?

• Composite Functions [Connie, 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)?

• Composite Functions Using Logarithms [Sitzer, 3/10/1996]
  Suppose f and g are functions defined by f(x) = x+2 and g(x) = x. Find all x > -2 for which: 3^[g(x)*logbase3 f(x)] = f(x).

• Composite Functions [Santoso, 4/5/1996]
  1) fog(x) = 7x + 3; gof(x) = 7x - 3; f(0) = 1; g(0) = .....

• Graphing Sin and Cosine Functions [BasuC, 5/24/1996]
  I really need some help in finding the graphs of the following functions...

• Functions Without a Second Derivative [Theta, 6/28/1996]
  What are some examples of functions of a real variable whose derivatives don't have derivatives?

• Functions of Imaginary Numbers [Pooh, 7/31/1996]
  Does (ln i) itself exist? Where does e^iA = cos A + i sin A come from?

• Inverse Function [Enzo, 8/28/1996]
  What is the inverse function of f(x) = 1 - 2x? What is the general method for finding an inverse function?

• Inverses of Functions [Senter, 9/2/1996]
  How do I find the inverses and determine whether the inverses are functions?

• Integrating X^x, Closed Form [Judes, 10/31/1996]
  How do you express the equation y = xcosx in terms of y?

• Proving One Function is Greater Than Another [Brenchi, 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)].

• Maximizing Output of a Restricted Function [Lovelace, 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.

• Finding the Roots of a Function [Thompson, 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

• Why Can't Some Functions be Integrated? [Shenoy, 11/06/1996]
  How do I evaluate Integral[x tanx dx]?

• Finding the Function [Bolibar, 12/17/1996]
  Given five related functions which have only one variable each, find the function that contains same all five variables.

• Quarter Circles [Cheung, 12/20/1996]
  Find a function that graphs a quarter of a circle in quadrants II and IV.

• Range of a Function [Tracy, 01/05/1997]
  Describe the range of the function g(x) = e^(-2x).

• Brackets or Parentheses? [McDonald, 01/07/1997]
  When using interval notation to describe when a function is increasing and decreasing, how do I know whether to use brackets or parentheses?

• One to One Correspondence Between Sets [Nicholas, 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, ...}?

• Naming and Graphing Functions [Perfect, 06/16/1997]
  Can you tell me about the different families of functions?

• Domain and Range [Jenni, 08/25/1997]
  Can you please explain about domain and range?

• Differentiating a Quotient [Gleghorn, 08/29/1997]
  Given that g(x) = x-4/x+3, find: {g (x+h) - g(x}/h).

• Closed Form Solutions [Batterman, 09/16/1997]
  What is the exact mathematical definition of a closed form solution?

• Finding the Domain of a Function [Breeding, 09/16/1997]
  How do you find the domain of this function: f(x) = (4+x)/(x^2-9) ?

• Express a Function in Terms of Another Function [Anes, 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.

• Derive the Function of Celsius to Fahrenheit [Larrandson, 09/22/1997]
  How do you get the equation of Celsius to Fahrenheit and vice versa? How would you graph that function?

• Is the Function Invertible? [Lanka, 09/22/1997]
  For the following functions f(x) decide if the funcion is invertible as a function from R to R...

• Implicit Functions [Norris, 11/26/1997]
  Please give me a definition and several examples of an implicit function.

• Find a Function, Integrals [Lanka, 12/17/1997]
  Suppose the graph of f has the formula f(x)=-x+1 for 0<=x<=1; x-1 for 1<x<=2 ...

• Inverse Function [Jaime, 01/02/1998]
  Show that f(x)= X/X-1 is its own inverse.

• Composite Functions [Nutty, 01/11/1998]
  My students can't understand composite functions.

• Odd And Even Functions [Ngchula, 01/11/1998]
  The differences between odd and even functions, and examples of each.

• Connecting the Dots [Freeman, 02/02/1998]
  How do you know whether or not to connect the dots when graphing a real-life function?

• One-to-one Function [Daley, 02/05/1998]
  If A = (0,1) and B = (a,b) show that there is a one-to-one function from A onto B.

• Root-Finding [Siddiqui, 02/09/1998]
  Given a polynomial of 3rd degree: X^3 + 5X - 2 = 0, show that this function has EXACTLY one root over the interval [0,1].

• How to Find the Range of a Function [Hunt, 02/25/1998]
  How do you find the range of a function like g(x) = (x+1)/(x^2-1)?

• Functions that are their own inverses [Gaab, 02/28/1998]
  Can a function ever be its own inverse?

• Domain and Range of an Inverse Function [Min, 03/18/1998]
  Why is the domain of a function equal to the range of the inverse function?

• Continuity and Differentiability of Piecewise Function [Hopson, 05/04/1998]
  What values of "a" and "b" will make this piecewise function continuous and differentiable?

• The Gamma Function and Its Derivative [Geib, 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!?

• What is the Gamma Function? What is Gamma of 4? [Masdeo, 05/28/1998]
  Deriving G(4) = 3! from the gamma function integral.

• Graph of Circle as a Function [Thompson, 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?

• Testing For Symmetry and Even/Odd Functions [Stevens, 06/08/1998]
  How do you test for the following: symmetry about the x-axis, symmetry about the y-axis, symmetry at the origin, and even or odd functions?

• Odd and Even Function Constructions [Brinson, 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.)?

• Hyperbolic Functions [Yoshi, 06/27/1998]
  What does the inverse of sinh or cosh mean?

• Sketching a Function [MatherWu, 07/06/1998]
  Can you help me piece a function together so that the following hold? It is increasing and concave up on (-infinity, 1) ...

• Summing Four Roots of an Even Function [Wu, 07/27/1998]
  If f(2+x) = f(2-x) and f(x) = 0 has exactly four distinct real roots, what is the sum of these roots?

• Manipulating Roots [Wave, 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.

• Inverse Functions [Ishmael, 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?

• An SAT Question on Functions [Rachel, 08/23/1998]
  Let the "tricate" of a number x be defined as one-third of the smallest multiple of 3 greater then x...

• Domain and Range of x^x [Larry, 08/24/1998]
  What is the domain and range of x^x?

• Translating Functions [Mosley, 08/27/1998]
  Find f(2+h), f(x+h), and f(x+h)-f(x)/h where h cannot = 0 for f(x) = x/(x + 1). Explain how the following graphs are obtained from the graph of y = f(x)...

• Functions: Domain, Range, and Piecewise [Antompietri, 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?

• Meromorphic Functions [Desabrais, 09/18/1998]
  What is a meromorphic function?

• Inverse Functions [Smith, 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?

• Greatest Integer Functions [Laura, 09/27/1998]
  Can you help me solve for the graph of [y]=[x], where [] is the greatest integer function?

• Domain, Range, and Asymptote [Summer, 10/12/1998]
  How do you find the domain, range, and asymptote of a function?

• Hyperbolic Functions [Weber, 10/14/1998]
  Can you explain hyperbolic functions? How are they defined? How do you evaluate them?

• Relations versus Functions [Nicole, 10/27/1998]
  What is the difference between a relation and a function? What about the domain and range of a function?

• Onto (Surjective) Functions [Taro, 11/03/1998]
  I don't understand onto functions. Would you please explain them in detail?

• The Phi Function [Browne, 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)?

• Composing Functions [Schoolcraft, 12/02/1998]
  I'm trying to find f-of-g where f(x) = 2x and g(x) = 3x^2 + 1. What happens when you compose two functions?

• Derivatives [Rakesh, 12/07/1998]
  Can you write x^2 = x+x+...+x (x times) if x is not a positive integer?

• O and o Functions [Kuhn, 01/08/1999]
  How do big O notation and little o notation show the relation between two functions?

• Symmetry Tests [Scardino, 01/12/1999]
  How can you know whether a graph is symmetric to the x-axis, y-axis, or the origin? What does the symmetry mean?

• The Function g(x) = gcd(x, 10) [Mejia, 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.

• Constructing Polynomials [Zhong, 01/20/1999]
  I need to find polynomials given certain properties. For example, find a polynomial P(x) that is of degree 4, where P(x) = P(-x), ...

• Inverse Functions [Knote, 02/11/1999]
  If y = arcsec(x) then what does x equal? How is this problem related to inverse functions?

• Range and Domain of a Graph [Eric, 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).

• Why use f(x)? [Ed , 02/14/1999]
  Why use f(x) in an equation instead of y?

• Natural Domain of a Function [Tara, 03/12/1999]
  Some inputs don't make sense for some functions.

• Horizontal and Vertical Asymptotes [Kristin, 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?

• Explicit Inverses [Yin, 04/21/1999]
  How do you solve x^x = 5 without graphing or guessing?

• Functions and Function Tables [Chris, 05/11/1999]
  I don't understand functions and function tables. Can you help me?

• Inverse Functions One-Way Only [Hewitt, 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?

• Expansion of (x+y)^(1/2) [Dheera, 06/07/1999]
  Is there a way to expand (x+y)^(1/2)? If so, how is it derived?

• Finding x for e^x = cos(x) [Lilley, 06/25/1999]
  How can I solve e^x = cos(x) for x?

• Composition of Functions [Pichelmann, 07/23/1999]
  How do I find f(g(x)) if f(x) = x+2 and g(x) = 3x-1?

• Equation for a W-Shaped Graph [Clarke, 08/01/1999]
  What sort of equation has a graph that is shaped like a 'W'?

• Determining whether a Function is Even or Odd [Lanning, 11/09/1999]
  How do you determine whether a function is an even function, an odd function, both, or neither?

• Inventing an Operation to Solve x^x = y [RedSting, 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?

• Newton-Raphson Method [Dyer, 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?

• Assigning Random Numbers [Bennett, 05/16/2000]
  I am using a programming language and have a random number generator that can generate a random number of 0, 1, or 2. How can I assign those three values to 4, 12, and 14?

• X^X^X^... = 2 [Dominic, 05/17/2000]
  How can I solve x^x^x^... = 2 for x?

• Taking the Natural Log of e^(ki) [Thomas, 05/18/2000]
  How is the natural log defined for e^(ki)? Applying the equation e^(i*2pi) = 1 we get ln[e^(i*2pi)] = ln[1], so i*2pi = 0, which doesn't seem possible.

• Formulas for N-Dimensional Spheres [Weeks, 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?

• Infinity as a Skolem Function [Zaba, 10/28/2000]
  Is infinity an absolute concept, a relative concept, or both?

• Chaotic Functions [Suzie, 10/30/2000]
  Can you give some mathematical examples of chaos theory?

• Functions O(k) and Omega(k) [Ganor, 01/20/2001]
  Please explain the difference between the functions O(k) and Omega(k).

• Circular Functions [Sheena, 01/27/2001]
  How do you define circular functions? Can you give me an example?

• Finding The Domain and Range of a Function [Armstrong, 01/27/2001]
  How do I find the domain and range of a function? My problem is f(x)= x squared -2.

• Raised Cosines [Deborah, 01/28/2001]
  What are raised cosines, and how do they work?

• Defining Transcendental Functions [Ruth, 02/05/2001]
  Can you give me a clear definition of "transcendental function"? How are algebraic, elementary, and transcendental functions related?

• Theta Notation: Complexity and the Step Function [Boxer, 02/05/2001]
  What does the Greek letter theta mean in this formula?

• Functions and Equations [Abby, 02/10/2001]
  When is an equation also a function? How do you recognize a function?

• Real and Rational Numbers [Eileen, 02/27/2001]
  How can I show that the number of rational numbers between 0 and 1 is the same as the number of natural numbers (considering the ordering of fractions: 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5...)?

• Continous Random Variables [Brian, 03/15/2001]
  Let c be a constant and consider the density function... find the value of c and the cumulative distribution function F(y); compute F(1); compute P(y greater than .5).

• Solving an Exponential Equation [Joe, 04/02/2001]
  Solve for x: x^3 = 2^x. I have created a computer program to solve this equation using the brute force method. x is close to 1.373468, but I can't isolate x.

• Big O Notation and Polynomials [Paul, 04/12/2001]
  Given the function f(x) = (x^3 - (4x^2) + 12)/(x^2 + 2), how can I find a polynomial function g(x) such that f(x) = O(g(x)) and g(x) = O(f(x))?

• What are Piecewise Functions? [stephanie, 05/13/2001]
  I know piecewise functions are based on expressions between specific intervals, but I do not know how to describe this function family.

• Composition Functions with Added x Value [Eileen, 05/13/2001]
  If x = 1, evaluate g(f(f(x))). I'm confused with this added value of x = 1.

• Inverses [Gator, 06/05/2001]
  What is an inverse?

• Solving by Interpolation [Rosly, 05/30/2001]
  Given y = 10 and b = 1.419, find X in the equation y = (b^(-0.25X)) + 0.84X.

• Are All Functions Equations? [Maureen, 07/16/2001]
  When my x's are not continuous, would I still have a function since the vertical line test might in fact not touch a point at all?

• ONTO and INTO [Pawntep, 07/27/2001]
  What is the difference between ONTO and INTO when you describe a function?

• Domain of a Composite Function [Michele, 08/30/2001]
  I don't completely understand how to find the domain of a composite function. Could you please explain it?

• Graphing Piecewise Functions [Mary, 09/09/2001]
  I do not understand how to graph piecewise functions.

• Graphing Limits [Justin, 09/12/2001]
  I was hoping you could explain the concept of "limits" and how to read them from graphs.

• Big O, Omega, and Sigma [Jeremy, 09/19/2001]
  I cannot understand how something can be both Big O and Omega (aka Big Theta). A general explanation of O/Omega/Theta would be helpful.

• Recognizing Functions [Miles, 09/30/2001]
  How do I know when there is a function on a graph?

• Reverse Modulus Operator [Charles, 10/09/2001]
  Is there an operator that would return 2 when we we do 6 * 0, * being this new operator?

• Floor and Mod Functions Discontinuous [Alkhaled, 10/10/2001]
  What is f(x) = x mod b for some fixed b?

• Graph with f(x) [Angela, 10/17/2001]
  Sketch the equations: y = f(x)+ 2; y = f(x-3); y = 2f(x).

• Fourier Transforms [Stuart, 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).

• Piecewise Function [Amie, 10/23/2001]
  How do you graph the piecewise function?

• 2^4 = 16 AND 4^2 = 16 [Yvonne, 10/29/2001]
  Can you think of any other pair of unequal numbers that share the same relation as 2 and 4 in the above example? What was your strategy?

• Graphing Reciprocal Functions [Willa, 11/01/2001]
  I know what a reciprocal is, but I don't know how to graph it.

• Degree of Constant Function [Masha, 11/08/2001]
  We think F(x) = 1x^0 is not a polynomial function (because polynomials shouldn't have discontinuities), but F(x) = 1 is a polynomial. And F(x) = 1 still has degree 0 but for reasons we can't explain.

• Exponentially Varying Functions [Eric, 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?

• Inner Product and L-2 Distance [Queenie, 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?

• Inverse Function [Trevor, 11/30/2001]
  I got this question wrong: y = (-5x - 2) / (-x + 1).

• Purpose of Absolute Value and Piecewise Functions [Kyle, 12/09/2001]
  What is the purpose of absolute value functions and of piecewise functions?

• Inverse Functions in Real Life [Denise, 01/25/2002]
  I would like to know if you have a good example of how inverse functions would be used in real life.

• Constant Function Zero and Orthogonal Functions [Neeraj, 02/08/2002]
  The constant function f(x) = 0 will give zero no matter what function g(x) it is integrated with. Does this mean that the constant function zero is orthogonal to all functions? Also, what could be the geometrical interpretation of orthogonal functions?

• Importance of Linear Functions [Abreeka, 02/06/2002]
  What is the importance of linear functions in the real world, and how are they used?

• 2^x = x^2 [Jeremy, 02/13/2002]
  Find the real value without graphing.

• Derivative of an Odd Function [Sheri, 02/27/2002]
  How do you show that the derivative of an odd function is even?

• Walsh Spectra [Laurie, 03/07/2002]
  What are Walsh spectra?

• e^pi vs. pi^e [Rajesh, 03/20/2002]
  Which is greater, e^pi or pi^e? I would like to have a simple proof.

• Rate of Change Constant? [Mariam, 03/20/2002]
  The rate of change of e^x is e^x. Does this mean that the rate of change is constant? Why are sinx,cosx,... and sinhx,coshx,.... similar?

• Range [Teresa, 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.