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.
- 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
- 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 W-Shaped Graph [08/01/1999]
What sort of equation has a graph that is shaped like a 'W'?
- Equation Solving with Newton-Raphson 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 b-th 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
- Everyday Applications of Inverse Functions [11/06/2011]
A student seeks real-world 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
- 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; x-1
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 y-value level off as the
x-value 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
- Finding the Domain of a Function [09/16/1997]
How do you find the domain of this function: f(x) = (4+x)/(x^2-9) ?
- 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 Non-Integers [01/06/2005]
I thought the factorial function was only defined for positive
integers, but I notice that my calulator returns a value for
non-integers 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^8-5x^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
- 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) = 2x-1 and g(x) = (x+5)/2.
- Fixed-Point 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 N-Dimensional 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 4-dimensional 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 Newton-Raphson Method and LambertW function.
- Functional Equation [02/18/2008]
If 2f(x) + f(1-x) = 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
- 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 X-Axis [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 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.