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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.

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?

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?

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