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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.
- 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.
- 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
- Inverses of Trigonometric Functions [08/07/2002]
How can the functions cosine and sine have inverses of arcsin and
- 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
- 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
- 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^2-1), 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 real-world 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
- 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, non-constant, 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
- 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?