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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.
 Domain, Range, and Translating Functions [06/29/2004]

Given the graph of y = f(x), domain [4,3], range [2,3], sketch other
functions, such as y = f(x) and y = f(x  2), and find the domain
and range of each.
 e^pi Greater Than pi^e [03/29/2003]

How can I show that e^pi is greater than pi^e without using a
calculator?
 e^pi vs. pi^e [03/20/2002]

Which is greater, e^pi or pi^e? I would like to have a simple proof.
 Equation for a WShaped Graph [08/01/1999]

What sort of equation has a graph that is shaped like a 'W'?
 Equation Solving with NewtonRaphson Method [07/23/2004]

Is there a way to solve F = A*(((1+i)^n)  1)/i for i, such that we have
i = ...? I'm thinking I need to use a logarithm, but I'm stuck.
 Equivalent Fractions, Exponential Implications [02/17/2011]

A student wonders how x^(a/b) changes for different multiples of a and b, and what difference it makes if you express that function as the bth root of x^a. Doctor Ali corroborates the student's suspicion that domains matter  as does the order of operations.
 Evaluating Composite Functions [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)?
 Everyday Applications of Inverse Functions [11/06/2011]

A student seeks realworld applications of inverse functions. Doctor Peterson starts
with an overview before suggesting several examples.
 Expansion of (x+y)^(1/2) [06/07/1999]

Is there a way to expand (x+y)^(1/2)? If so, how is it derived?
 Explaining Independent and Dependent Variables [10/25/2002]

Can you elaborate on the definitions of independent and dependent
variables?
 Explicit Inverses [04/21/1999]

How do you solve x^x = 5 without graphing or guessing?
 Exponentially Varying Functions [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?
 Express a Function in Terms of Another Function [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.
 Fazed by Fourier Transform Phases [11/14/2011]

Complex output from an inverse discrete Fourier transform (DFT) raises doubts: would
discarding imaginary values mean losing important information? Doctor Douglas
explains why the phase of each Fourier component matters.
 Find a Function, Integrals [12/17/1997]

Suppose the graph of f has the formula f(x)=x+1 for 0<=x<=1; x1
for 1<x<=2 ...
 Finding a Formula to Fit Data Points [06/23/2004]

How can I find a formula that contains the points (500,150),
(1000,200), and (1500,225) and has the yvalue level off as the
xvalue gets larger?
 Finding a Limit by Iterating a Function [04/21/2004]

Given f(x) = 2x/(x + 1), determine the limit as n approaches infinity
of f^n(x) for x > 0.
 Finding an Inverse Function [8/28/1996]

What is the inverse function of f(x) = 1  2x? What is the general method
for finding an inverse function?
 Finding the Arc Length of a Hanging Catenary [07/23/1997]

A catenary is suspended between two equal poles 400 feet apart at equal
height; it sags in the center 40 feet...
 Finding The Domain and Range of a Function [01/27/2001]

How do I find the domain and range of a function? My problem is f(x) = x
squared 2.
 Finding the Domain of a Function [09/16/1997]

How do you find the domain of this function: f(x) = (4+x)/(x^29) ?
 Finding the Domain of a Function [02/10/2009]

I'm having trouble finding the domain of a function. What is the
domain and how does it work?
 Finding the Factorial of NonIntegers [01/06/2005]

I thought the factorial function was only defined for positive
integers, but I notice that my calulator returns a value for
nonintegers like 3.5! What's going on?
 Finding the Function [12/17/1996]

Given five related functions which have only one variable each, find the
function that contains same all five variables.
 Finding the Inverse of y = x^x [08/31/2006]

I have been trying to find the inverse function for y = x^x and once I
exchange x and y I can't find a way to solve for y. Can you help?
 Finding the Roots of a Function [11/4/1996]

How do you find the positive roots of x^9+3x^85x^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
 Finding x for e^x = cos(x) [06/25/1999]

How can I solve e^x = cos(x) for x?
 Find the Function (fog)(x) [07/02/2003]

Let f(x) = 2x1 and g(x) = (x+5)/2.
 FixedPoint Theorem [05/02/2002]

An intuitive reason why the theorem is true, and a proof.
 Floor and Mod Functions Discontinuous [10/10/2001]

What is f(x) = x mod b for some fixed b?
 Formulas for NDimensional Spheres [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 4dimensional sphere?
 Fourier Transforms [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).
 Fractional Bases in Exponential and Logarithmic Functions [04/06/2009]

In the equation log_a(x) = a^x, can the base be a fraction? Doctor Ali
introduces a student to the NewtonRaphson Method and LambertW function.
 Functional Equation [02/18/2008]

If 2f(x) + f(1x) = x^2 for all x, then f(x) = ?
 Functional Transformations Explained and Undone [09/09/2013]

A student wonders why horizontal transformations behave "opposite" to their original functions. Doctor Peterson explains with one example shifted by addition, and another scaled by
multiplication.
 Function? Check. Inverse? Check. Composition Order? Check the Domains [10/17/2015]

An adult seeks a function that, when composed with its inverse, does not equal the
inverse's composition with its function. Emphasizing domains and ranges,
Doctor Peterson obliges with some mismatching examples.
 Function Crossing XAxis [07/11/2003]

Find the condition that must be satisfied by k in order so that the
expression 2x^2 + 6x + 1 + k(x^2 + 2) may be positive for all real
values of x.
 Function Deduction from Composition [11/19/2013]

Given one function and the result of its composition with a second function, a student
wonders how to recover the second function. Doctor Peterson shows how substitution
solves such problems; Doctor Rick follows up with a method that relies on the inverse
function.
 Function Exponentiation Convention [04/21/2013]

A student seeks clarity on the meaning of f^2(x) and other expressions that juxtapose
functions and exponents. Doctor Vogler empathizes, outlining the differing
interpretations, before recommending contextual clues — and pointing up a
larger lesson about ambiguity.
 The Function g(x) = gcd(x, 10) [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.
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