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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 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 g)?

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 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; 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 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^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 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) = 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 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 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.

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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