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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.
 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 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.
 Function Its Own Inverse [01/02/1998]

Show that f(x) = X/X1 is its own inverse.
 Functions [11/12/1994]

I'm a first year algebra student having difficulty learning about
functions. Exactly how do they work?
 Functions and Equations [02/10/2001]

When is an equation also a function? How do you recognize a function?
 Functions and Function Tables [05/11/1999]

I don't understand functions and function tables. Can you help me?
 Functions and Inverses [05/29/2002]

Find sets A, B, and C, and functions f:A>B and g:B>C, such that
(g o f) is both injective and surjective, but f is not surjective
and g is not injective.
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