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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.

All About Functions [11/06/1996]
Could you please explain functions?

Defining 'Undefined' [09/15/2003]
If a function is 'undefined at x', does this refer only to vertical asymptotes, or to other discontinuities as well?

Definition of the Signum Function [05/31/2000]
Can you give me a simple definition of the signum function, and any practical examples of its usage?

Exp, Log, and Ln Functions Explained [7/31/1996]
What is the exp function? When is it needed? Also, how do I calculate Log and Ln functions with basic arithmetic and logic?

Function Machine [10/26/1996]
How do you find the domain and range of a function?

Function Tests [02/19/1997]
What is the reasoning behind the vertical and horizontal line tests?

Interval Notation [4/1/1996]
I need to learn about interval notation in terms of domain and ranges.

Mapping Functions in the Real World [3/20/1995]
What is the purpose of learning to map a function? What is it used for in the real world?

Rational Inequality [10/09/2001]
Solve this rational inequality and give an answer in interval notation: - 5/(3h+2) greater than or equal to 5/h.

Sometimes, Always, or Never True? [02/12/2002]
Is this statement always, sometimes, or never true: f(g(x))=g(f(x)) ?

What Are Quadratic Functions? [02/27/2003]
What is the difference between a quadratic function and a quadratic formula?

What is a Function? [06/14/2001]
I've read many definitions and I've asked many teachers, but I still don't completely understand.

Why is Zero the Limit? [02/25/2002]
Why is zero called the limit of the terms in the sequence the limit of 1 over n, as n approaches infinity, equals zero?

x Factorial and the Gamma Function [05/29/1998]
What is x! when x is 0, negative, or not a whole number?

2^4 = 16 AND 4^2 = 16 [10/29/2001]
Can you think of any other pair of unequal numbers that share the same relation as 2 and 4 in the above example? What was your strategy?

2^x = x^2 [02/13/2002]
Find the real value without graphing.

ABS(f(x)) and f(ABS(x)) Graphs [08/27/2013]
A student seeks help graphing functions transformed by absolute values variously acting on the whole function, and on just the argument, by itself. Doctor Peterson talks through sketching and checking such transformations.

Absolute Value and Continuity of Functions [09/15/2004]
I know that the absolute value of a continuous function is also continuous. Is the opposite true? That is, if the absolute value of a function is continuous, is the function continuous?

Algebraically Equivalent Functions [06/27/2002]
If a function can be manipulated so that it can't have a denominator equal to zero (and thus be undefined for that value), why is the original function still considered undefined at that value?

Approaching Zero and Losing the Plot [11/11/2010]
Looking near the origin at plots of y = x^n for ever tinier n, a student wonders why y = x^0 does not equal zero. By emphasizing two different limits, Doctor Ali gets the student back into line -- specifically, y = 1.

Approximating f(x - 1) in Terms of Its First and Second Derivatives [08/15/2012]
A student seeks help verifying that f(x - 1) = f(x) - f'(x) + (1/2)f''(e). Doctor Schwa renames the function and applies Taylor's expansion to clarify where to substitute, providing just the right boost.

Are All Functions Equations? [07/16/2001]
When my x's are not continuous, would I still have a function since the vertical line test might in fact not touch a point at all?

Assigning Random Numbers [05/16/2000]
I am using a programming language and have a random number generator that can generate a random number of 0, 1, or 2. How can I assign those three values to 4, 12, and 14?

Asymptote of a Function [06/02/2002]
Determine the value of A so that y = (Ax+5)/(3-6x) has a horizontal asymptote at y = -2/3.

Big O Notation and Polynomials [04/12/2001]
Given the function f(x) = (x^3 - (4x^2) + 12)/(x^2 + 2), how can I find a polynomial function g(x) such that f(x) = O(g(x)) and g(x) = O(f(x))?

Big O, Omega, and Sigma [09/19/2001]
I cannot understand how something can be both Big O and Omega (aka Big Theta). A general explanation of O/Omega/Theta would be helpful.

Brackets or Parentheses? [01/07/1997]
When using interval notation to describe when a function is increasing and decreasing, how do I know whether to use brackets or parentheses?

Calculus of Piecewise Functions [06/07/2003]
Can I take the integral or derivative of a piecewise function like the floor function [u] or the absolute value function |u| and still notate it in concise form, |U| or [U]?

Can f'(-1) Equal Zero and f''(-1) Not Equal Zero? [03/23/2004]
Is it possible to have a derivative of zero and then have a double derivative that is not zero at that same x value? How?

Cases Where the Newton-Raphson Method Fails [06/30/2005]
Why does the Newton-Raphson method work for some functions but not for others?

Catenary Curve [03/30/1999]
Find the vertex of a catenary curve.

Chaotic Functions [10/30/2000]
Can you give some mathematical examples of chaos theory?

Circular Functions [01/27/2001]
How do you define circular functions? Can you give me an example?

Closed Form Solutions [09/16/1997]
What is the exact mathematical definition of a closed form solution?

Coconuts, Forwards and Backwards [02/02/2010]
Doctor Greenie answers a chestnut about repeated division and remainders, first working the question forwards before using the inverse of a function to solve the same problem backwards much more easily.

Composing Functions [12/02/1998]
I'm trying to find f-of-g where f(x) = 2x and g(x) = 3x^2 + 1. What happens when you compose two functions?

Composite Functions [4/5/1996]
1) fog(x) = 7x + 3; gof(x) = 7x - 3; f(0) = 1; g(0) = .....

Composite Functions [01/11/1998]
My students can't understand composite functions.

Composite Functions Using Logarithms [3/10/1996]
Suppose f and g are functions defined by f(x) = x+2 and g(x) = x. Find all x > -2 for which: 3^[g(x)*logbase3 f(x)] = f(x).

Composition Functions with Added x Value [05/13/2001]
If x = 1, evaluate g(f(f(x))). I'm confused with this added value of x = 1.

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