See also the
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.
- 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...)?
- Reasoning Reversibly? [01/31/2017]
If we conclude that a proposition and its converse both are true, what if anything does
that say about the reversibility of irreversible operations that we used to reach those
conclusions? By distinguishing between operations and their results, Doctor Peterson
straightens out the chains of reasoning.
- 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?
- 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 ,
- 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
- 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)) +
- 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, so i*2pi = 0, which doesn't
- 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.
- A Transformational Tour [01/28/2017]
Given the graph of a transformation, how do you recover the original function? As he
dissects reflections, dilations, and shifts, Doctor Peterson shares observations about
what direct inspection reveals — and what to expect from textbook problems.
- 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
- 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 =