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Browse High School Functions
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Selected answers to common questions:
Composition of functions.
Domain and range.
Inverse of a function.
- 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
- 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.
- 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
- 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
- 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 =
- Composition of Functions [07/23/1999]
How do I find f(g(x)) if f(x) = x+2 and g(x) = 3x-1?