Ask Dr. Math

High School Archive

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.

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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.

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.

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.

Composition of Functions [07/23/1999]

How do I find f(g(x)) if f(x) = x+2 and g(x) = 3x-1?

Connecting the Dots [02/02/1998]

How do you know whether or not to connect the dots when graphing a real- life function?

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