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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.



Inverse Functions One-Way Only [06/01/1999]
Are there two functions, f and g, such that (f of g)(x) = x but (g of f)(x) does not equal x?

Inverse Functions - Which Statement is True? [09/20/1998]
Does (f o g)^(-1) equal f^(-1) o g^(-1) or g^(-1) o f^(-1)? How can you prove it?

Inverse of a Function vs. Inverse Proportionality [11/20/2002]
Finding the inverse of a function and graphing yields a graph that has been reflected in the line y = x relative to the function. Inverse proportionality, however, yields a reciprocal relation graphically. Why do these two things have similar names yet mean different things?

Inverse of a Multivariate Function [05/30/2002]
Let f:NxN -> N such that f(x,y) = 2^x(2y + 1) - 1 for all natural numbers x, y. Let the inverse of f, g be given by g:N -> NxN. Find the inverse of the function g.

Inverse of arg(z) [10/10/2003]
What is the inverse of the function arg(z)?

The Inverse of the Absolute Value Function [12/12/2008]
What is the inverse of the absolute value function? I know the range must be restricted if the inverse is to be a function.

Inverses [06/05/2001]
What is an inverse?

Inverses and Reciprocals of Functions [09/28/2005]
What is the difference between f^(-1)(x) and f(x)^(-1)?

Inverses of Trigonometric Functions [08/07/2002]
How can the functions cosine and sine have inverses of arcsin and arccos respectively?

Inverse Statements [08/20/1998]
Say there are 3 + 4 times as many girls as boys. Is this the same as saying there are -3 + 1/4 as many boys as girls?

Inverse Trigonometric Ratios [07/01/2003]
Why aren't the inverse trigonometric ratios equal to 1/(the ratio), as is the case for numbers?

Invertible Functions [06/06/2003]
Consider as a function from R -> R (Real) and say whether the function is invertible: h(x) = (sgn x)* sqrt(abs(x)) where sgn is +1 if x is positive, -1 if x is negative, and 0 if x is 0.

Inverting Functions [07/19/2002]
To find the inverse of a function y=f(x), do I interchange the variables x and y, or do I solve for x in terms of y?

Irregular Sinusoidal Curves [08/24/2004]
Can you create a sinusoidal function with a fixed period such that the time span between the max and min is not the same as the time span between the min and max?

Is the Function Invertible? [09/22/1997]
For the following functions f(x) decide if the function is invertible as a function from R to R...

Is the Inverse a Function? [9/2/1996]
How do I find the inverses and determine whether the inverses are functions?

An Iterative Method of Calculating Pi [06/09/2004]
I recently saw a method of calculating pi that involves an iterative function, P(n + 1) = P(n) + sin(P(n)) where P(n) is the approximation of pi at the nth iteration.

Linear Equations and Standard Form [04/04/2002]
What do the variables mean in the standard form of a linear equation (ax+by=c)?

Linearity and Concavity [12/07/2003]
Why does a linear function have no concavity?

Line Crossing Horizontal and Vertical Asymptotes [03/16/1999]
For the function y = (5x+1)/(x^2-1), why does the line between its two vertical asymptotes and one horizontal asymptote cross?

Lipschitz Continuous Functions [04/27/2008]
Show that a Lipschitz continuous function is uniformly continuous on a subset and that the converse is not necessarily true. Also, give an example to show that Lipschitz continuous functions are not necessarily differentiable.

Logs of Complex Numbers [02/11/2004]
Give an example showing that Log(z1/z2) does not equal Log(z1) - Log(z2) where z1 and z2 are complex numbers.

Manipulating Roots [08/08/1998]
If a and b are the roots of the equation 3x^2 - 5x + 1 = 0, find the equation whose roots are a/b and b/a.

Max and Min of Functions without Derivative [04/22/2003]
Is there a general way to find the maximum and minimum of cubic functions without using derivatives?

Maximizing Output of a Restricted Function [11/1/1996]
Create a function whose domain is restricted to complex numbers but whose range is real, that is, non-constant, has no constant term, and contains no number greater than 3.

Measurable Functions [09/13/2004]
Can you provide some motivation for the definition of a measurable function?

Meromorphic Functions [09/18/1998]
What is a meromorphic function?

The Meteorologists' Theorem [1/6/1995]
Prove the "Meteorologists' Theorem": At any given moment, there are two diametrically opposite points on the (spherical?) Earth's surface where the temperatures are equal and the barometric pressure are equal.

Multiplying Square Roots of Negative Numbers [11/25/2003]
It seems to me there are two possible ways to interpret a problem like sqrt(-2) * sqrt(-2). One way I get 2 and the other way I get -2. Which solution is correct? What's wrong with the other one?

Mystery Operation: What Do These Strange Symbols Mean? [03/09/2010]
How do you evaluate the composition of two functions when you don't even recognize their operational symbols? Doctor Ian unravels the mystery behind weird symbols, explains their instructional purpose, and uses grouping parentheses to show how to compose functions according to the order of operations.

Naming and Graphing Functions [06/16/1997]
Can you tell me about the different families of functions?

Natural Domain of a Function [03/12/1999]
Some inputs don't make sense for some functions.

Newton-Raphson Method [02/28/2000]
How can I find the 1st, 2nd, 3rd iterations and the parameters of x^3 - 13.1x^2 + 48.48x - 46.62 using the Newton-Raphson method?

Newton-Raphson Method [06/24/2009]
Are there any equations that cannot be solved using the Newton-Raphson method, regardless of the initial estimate?

Newton's Method for Finding Roots of Functions [06/03/2003]
Is there an easy way to learn and understand this method or an example that would allow me to understand the steps taken to work through the problem?

Nondifferentiable Functions [11/6/1994]
In my calculus book, it mentions a function that is not differentiable at any point due to the fact that it is not smooth at any point. It does not go any farther, and I was interested in hearing more.

Normalizing Ranges of Numbers [04/22/2002]
How do I convert ranges like 5-200 to a scale of 1-10?

O and o Functions [01/08/1999]
How do big O notation and little o notation show the relation between two functions?

Odd and Even Function Constructions [06/08/1998]
If f is an odd function and g is an even function, how do we combine the two in various constructions (adding, multiplying, etc.)?

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

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