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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.

Recurrence Relation Resolution [11/25/2013]
A student struggles to determine the limit points and explicit formula for a recurrence relation complicated by powers and other operations. Exploiting derivatives and ratios, Doctor Vogler shows the way.

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).

Reversal of Function [08/25/2014]
A teen wonders if there is ever a time when x is a function of y. Doctor Peterson examines the conventional notation, then anticipates several related topics, with examples throughout.

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?

Shifting Images [10/21/2014]
A student wonders why transforming functions sometimes changes their image points, and other times does not. Inspecting the student's example functions more closely, Doctor Peterson turns the attention to the transformations, themselves.

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?

Square Root Function: Why Restrict Its Range to Non-Negative Numbers? [01/26/2010]
Doctor Peterson makes the case for non-negative principal roots.

Stretching Definitions, and Compressing [08/04/2014]
Given counterintuitive definitions in his textbook, a teen seeks clarity around how to describe the effect of the positive factor c in y = f(cx). Surveying other usages of "compress" and "stretch" around the web, Doctor Peterson turns up inconsistencies.

Summing Four Roots of an Even Function [07/27/1998]
If f(2+x) = f(2-x) and f(x) = 0 has exactly four distinct real roots, what is the sum of these roots?

Symmetry Tests [01/12/1999]
How can you know whether a graph is symmetric to the x-axis, y-axis, or the origin? What does the symmetry mean?

Taking the Natural Log of e^(ki) [05/18/2000]
How is the natural log defined for e^(ki)? Applying the equation e^(i*2pi) = 1 we get ln[e^(i*2pi)] = ln[1], so i*2pi = 0, which doesn't seem possible.

Taking the Partial Derivative of a Function [09/06/2002]
Are the left and right sides of an equation always symmetric?

Testing For Symmetry and Even/Odd Functions [06/08/1998]
How do you test for the following: symmetry about the x-axis, symmetry about the y-axis, symmetry at the origin, and even or odd functions?

Theta Notation: Complexity and the Step Function [02/05/2001]
What does the Greek letter theta mean in this formula?

Times that Call for Line Graphs? [11/26/2012]
A teacher wonders whether line graphs suit cumulative temporal data. Doctor Peterson talks through how to decide on the right representation, dispelling some rigid notions along the way.

To Invert Functions, First Subvert Routine [12/09/2010]
Don't you invert a function by just flipping its unknowns? Emphasizing a function as a relationship, and distinguishing variable names from their roles, Doctor Peterson clears up a misunderstanding borne of habitual exposure to canonical "y = f(x) ..." form and x as the independent variable on the horizontal axis.

Transforming Tables when f(x) Becomes f(abs(x)) [09/19/2013]
A student struggles with a parent function transformed by absolute value. Picking up on several of the student's insights, Doctor Peterson steps through an example.

Translating Functions [08/27/1998]
Find f(2+h), f(x+h), and f(x+h)-f(x)/h where h cannot = 0 for f(x) = x/(x + 1). Explain how the following graphs are obtained from the graph of y = f(x)...

Undefined and Indeterminable ... at the Same Time? [09/05/2010]
A student wonders whether the labels "undefined" and "indeterminate form" could apply to one and the same expression. Doctor Vogler considers several expressions, functions, and limits to distinguish the different contexts that call for such terminology.

Unraveling an Inverse Function [11/30/2001]
I got this question wrong: y = (-5x - 2) / (-x + 1).

Using Phase Shifts to Modify a Sine Curve [06/30/2005]
What modifiers can be used to fatten a sine wave while not affecting the zeros and peaks?

The Vertical Line Test for Functions [05/06/2004]
I'm having a problem understanding functions. I understand that for every "x" there must be only one "y". But the vertical line test is what is confusing me. Can you explain it to me?

Walsh Spectra [03/07/2002]
What are Walsh spectra?

What are Piecewise Functions? [05/13/2001]
I know piecewise functions are based on expressions between specific intervals, but I do not know how to describe this function family.

What is a Functional Transformation? [08/11/2002]
I would like a brief definition of a functional transformation and what an application is in early Calculus.

What is the Gamma Function? What is Gamma of 4? [05/28/1998]
Deriving G(4) = 3! from the gamma function integral.

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