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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.



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) = ?

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.

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/X-1 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.

Functions: Domain, Range, and Piecewise [08/31/1998]
What are piecewise functions? What are open and closed points? How do you figure out the domain and range of a function without graphing it on a calculator?

Functions of a Complex Variable [10/05/2002]
Show that w=f(z*), where z*=x-iy and f is a differentiable function, is not an analytic function of z.

Functions of Imaginary Numbers [7/31/1996]
Does (ln i) itself exist? Where does e^iA = cos A + i sin A come from?

Functions O(k) and Omega(k) [01/20/2001]
Please explain the difference between the functions O(k) and Omega(k).

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