See also the
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.
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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 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
- 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.