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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.

Odd And Even Functions [01/11/1998]
The differences between odd and even functions, and examples of each.

One to One Correspondence Between Sets [04/04/1997]
What is the one-to-one correspondence between the set of natural numbers and the set "S" = {1/3, 1/6, 1/12, 1/24, 1/48, ...}?

One-to-one Function [02/05/1998]
If A = (0,1) and B = (a,b) show that there is a one-to-one function from A onto B.

ONTO and INTO [07/27/2001]
What is the difference between ONTO and INTO when you describe a function?

Onto (Surjective) Functions [11/03/1998]
I don't understand onto functions. Would you please explain them in detail?

Order of Transformations of a Function [11/30/2005]
In what order would I perform transformations such as -f(x), f(-x), af(x), f(ax), f(x)+a, and f(x+a) if two or more are applied to f(x)? If I had f(ax+b), would I do the translation or the stretch first?

Parameter vs. Constant [12/08/2002]
Can you explain the difference between parameter and constant?

Parameter vs. Temperature Formula [10/28/2002]
I am trying to figure out a formula for frequency vs. temperature.

The Phi Function [11/21/1998]
What are the conditions on n,m so that phi(n*m) = phi(n)*phi(m)? What is phi(p^n*q^m)?

Piecewise Function [10/23/2001]
How do you graph the piecewise function?

Piecewise Functions on TI Graphing Calculators [02/08/2005]
I was wondering if you could tell me how to graph a piecewise function on a TI-83 plus graphing calculator?

Polar Coordinates and Logarithmic Spirals [04/25/2004]
The polar coordinate graph of a logarithmic spiral suggests that for a given angle, there is more than one possible r-value, since the graph continues to spiral around itself. How is it possible for an angle to be associated with more than one r?

Possible Values of f(0) [04/19/2002]
If f(x+y) = f(x)f(y), what is (are) the possible value(s) of f(0)?

Product Notation [10/26/2000]
Can the pi symbol also be used to mean multiplying all the terms in a series together?

A Project on Cycloids [01/16/1999]
Can you explain cycloids? How do you work with the parametric equations? What are their properties? How are they related to time?

Proving One Function is Greater Than Another [11/1/1996]
Prove that if n is a positive integer and x > 0, then x^n + 1/(x^n) > x^(n-1) + 1/[x^(n-1)].

Proving Subsets [07/01/2003]
Given the function f:X --> Y and subsets A of X and B of Y, prove the following statements: A is a subset of f^{-1}(f(A)); f(f^{-1}(B)) is a subset of B.

PSAT Math Question on Functions [10/17/2003]
For all values of r, let *r be defined as *r = (r + 2)/2. If *4 = x, then *x = ?

Purpose of Absolute Value and Piecewise Functions [12/09/2001]
What is the purpose of absolute value functions and of piecewise functions?

Quarter Circles [12/20/1996]
Find a function that graphs a quarter of a circle in quadrants II and IV.

Raised Cosines [01/28/2001]
What are raised cosines, and how do they work?

Range [04/02/2002]
If f(x) = x-3/x+3, I know the domain to be all Reals except for -3. The range, however, is all Reals except for 1. I don't know how they get the 1 algebraically.

Range and Codomain of a Relation [03/27/2004]
What is the difference between the range and the codomian of a relation?

Range and Domain of a Graph [02/12/1999]
Determine the domain and range of the graph of "f" that starts at (- 2,3), goes down to (0,0), and ends at (3,4).

Range of a Function [01/05/1997]
Describe the range of the function g(x) = e^(-2x).

Rate of Change Constant? [03/20/2002]
The rate of change of e^x is e^x. Does this mean that the rate of change is constant? Why are sinx,cosx,... and sinhx,coshx,.... similar?

Rational Function Range-Finding, with and without Calculus [06/01/2012]
A student seeks an analytic method for determining the range of (x^2)/(x + 1). Reinforcing that no one technique exists for all such rational functions with polynomial degree in their numerator greater than that in their denominator, Doctor Peterson outlines three approaches for the specific problem presented.

Real and Rational Numbers [02/27/2001]
How can I show that the number of rational numbers between 0 and 1 is the same as the number of natural numbers (considering the ordering of fractions: 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5...)?

Recognizing a Function From a Graph [09/03/2003]
How do you know whether a graph represents a function?

Recognizing Functions [09/30/2001]
How do I know when there is a function on a graph?

Relations in Real Life [10/02/2012]
A student seeks examples of functions in the real world. Doctor Ian gives her some instances of everyday conventions that feature restricted domains and ranges.

Relations versus Functions [10/27/1998]
What is the difference between a relation and a function? What about the domain and range of a function?

Remainder Theorem and Synthetic Substitution [06/17/2004]
Given P(x) = x^15 - 2004x^14 + 2004x^13 - ... - 2004x^2 + 2004x , calculate P(2003).

Reverse Modulus Operator [10/09/2001]
Is there an operator that would return 2 when we we do 6 * 0, * being this new operator?

Root-Finding [02/09/1998]
Given a polynomial of 3rd degree: X^3 + 5X - 2 = 0, show that this function has EXACTLY one root over the interval [0,1].

Root-Mean-Square [06/15/2003]
How can I visualize the distance between two functions?

An SAT Question on Functions [08/23/1998]
Let the "tricate" of a number x be defined as one-third of the smallest multiple of 3 greater then x...

Sequence of Integers [08/12/2008]
Find all functions f such that for each n in Z+ we have f(n) > 1 and f(n + 3)f(n + 2) = f(n + 1) + f(n) + 18.

Shifting Graphs [08/26/2002]
Why is the graph of y=f(x-c) just the graph of y=f(x) shifted c units to the right?

Simplifying and Working with Imaginary Numbers [04/11/2008]
What is the rule for simplifying an expression like sqrt(50)/sqrt(-5)? Do you get i*sqrt(10) or -i*sqrt(10)? Is there a general rule for simplifying imaginary square roots with regard to handling the i?

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