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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.
 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?
 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?
 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 OneWay 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
prove it?
 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.
 Inverse of arg(z) [10/10/2003]

What is the inverse of the function arg(z)?
 The Inverse of the Absolute Value Function [12/12/2008]

What is the inverse of the absolute value function? I know the range
must be restricted if the inverse is to be a function.
 Inverses [06/05/2001]

What is an inverse?
 Inverses and Reciprocals of Functions [09/28/2005]

What is the difference between f^(1)(x) and f(x)^(1)?
 Inverses of Functions, and Inverse Functions [02/12/2014]

A student's textbooks give her conflicting definitions about functions and their
inverses. Doctor Peterson makes sense of the disagreements by providing the bigger
picture.
 Inverses of Trigonometric Functions [08/07/2002]

How can the functions cosine and sine have inverses of arcsin and
arccos respectively?
 Inverse Statements [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?
 Inverse Trigonometric Ratios [07/01/2003]

Why aren't the inverse trigonometric ratios equal to 1/(the ratio),
as is the case for numbers?
 Invertible Functions [06/06/2003]

Consider as a function from R > R (Real) and say whether the function
is invertible: h(x) = (sgn x)* sqrt(abs(x)) where sgn is +1 if x is
positive, 1 if x is negative, and 0 if x is 0.
 Inverting Functions [07/19/2002]

To find the inverse of a function y=f(x), do I interchange the
variables x and y, or do I solve for x in terms of y?
 Inverting, Subverted [05/03/2015]

A teen wonders why interchanging a function's variables does not lead to its inverse.
With several examples and two different approaches, Doctor Peterson disambiguates
some commonly conflated notions.
 Irregular Sinusoidal Curves [08/24/2004]

Can you create a sinusoidal function with a fixed period such that the
time span between the max and min is not the same as the time span
between the min and max?
 Is the Function Invertible? [09/22/1997]

For the following functions f(x) decide if the function is invertible as a
function from R to R...
 Is the Inverse a Function? [9/2/1996]

How do I find the inverses and determine whether the inverses are
functions?
 An Iterative Method of Calculating Pi [06/09/2004]

I recently saw a method of calculating pi that involves an iterative
function, P(n + 1) = P(n) + sin(P(n)) where P(n) is the approximation
of pi at the nth iteration.
 Linear Equations and Standard Form [04/04/2002]

What do the variables mean in the standard form of a linear equation
(ax+by=c)?
 Linearity and Concavity [12/07/2003]

Why does a linear function have no concavity?
 Line Crossing Horizontal and Vertical Asymptotes [03/16/1999]

For the function y = (5x+1)/(x^21), why does the line between its two
vertical asymptotes and one horizontal asymptote cross?
 Line of Best Fit, and Flight [02/04/2015]

What do "extrapolation" and "interpolation" mean? What are their purpose? Doctor
Peterson defines the terms, then gives realworld examples based on tracking the
flight of his son's recent airplane trip.
 Lipschitz Continuous Functions [04/27/2008]

Show that a Lipschitz continuous function is uniformly continuous on a
subset and that the converse is not necessarily true. Also, give an
example to show that Lipschitz continuous functions are not
necessarily differentiable.
 Logs of Complex Numbers [02/11/2004]

Give an example showing that Log(z1/z2) does not equal Log(z1) 
Log(z2) where z1 and z2 are complex numbers.
 Manipulating Roots [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.
 Max and Min of Functions without Derivative [04/22/2003]

Is there a general way to find the maximum and minimum of cubic
functions without using derivatives?
 Maximizing Output of a Restricted Function [11/1/1996]

Create a function whose domain is restricted to complex numbers but whose
range is real, that is, nonconstant, has no constant term, and contains
no number greater than 3.
 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?
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