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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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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?

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 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?

Sketching a Function [07/06/1998]

Can you help me piece a function together so that the following hold? It is increasing and concave up on (-infinity, 1) ...

Sketching a Graph Given Information about Its Derivatives [07/30/2005]

We've been learning how to analyze a function by using the first and second derivatives to test if the graph is increasing/decreasing and concave up/down. But now we have to sketch the graph given some information about the derivatives and some specific points on the graph.

Sketching a Polynomial [04/04/2002]

Why should a curve change its position or sign at roots? Can't a curve have a positive value for all roots?

Sketching a Signal Graph Based on a Step Function [08/03/2006]

Sketch the graph of x(t) = 3(t+3)u(t+3) - 6tu(t) + 3(t-3)u(t-3) where u(t) is a step function.

Solving an Exponential Equation [04/02/2001]

Solve for x: x^3 = 2^x. I have created a computer program to solve this equation using the brute force method. x is close to 1.373468, but I can't isolate x.

Solving by Interpolation [05/30/2001]

Given y = 10 and b = 1.419, find X in the equation y = (b^(-0.25X)) + 0.84X.

Square Root Functions and Transformations [09/07/2009]

How can we tell by looking at the graph of a square root function if it is being horizontally compressed or vertically stretched?

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