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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.

Composition of Functions [07/23/1999]
How do I find f(g(x)) if f(x) = x+2 and g(x) = 3x-1?

Composition of Functions: Inequalities in the Domain, Inaccuracy in the Book [02/19/2013]
A student wonders why her textbook disagrees with her solution to the domain of a composite function. Doctor Peterson walks through three approaches, as well as a method that checks her answer — and the book's.

Connecting the Dots [02/02/1998]
How do you know whether or not to connect the dots when graphing a real- life function?

Constant Function Zero and Orthogonal Functions [02/08/2002]
The constant function f(x) = 0 will give zero no matter what function g(x) it is integrated with. Does this mean that the constant function zero is orthogonal to all functions? Also, what could be the geometrical interpretation of orthogonal functions?

Constructing Polynomials [01/20/1999]
I need to find polynomials given certain properties. For example, find a polynomial P(x) that is of degree 4, where P(x) = P(-x), ...

Continous Random Variables [03/15/2001]
Let c be a constant and consider the density function... find the value of c and the cumulative distribution function F(y); compute F(1); compute P(y greater than .5).

Continuity and Differentiability of Piecewise Function [05/04/1998]
What values of "a" and "b" will make this piecewise function continuous and differentiable?

Continuity of f(x) = sin(x)/x at x = 0 [01/20/2004]
Given f(x) = (sinx)/(x) for x =/= 0 and 1 for x = 0, show that f(x) is continuous at x = 0.

A Continuous Function on a Specified Interval [11/13/2003]
Does there exist a partition of the set X = [0,1) into two subsets A and B and a continuous function f(x) such that x in A ==> f(x) in B and x in B ==> f(x) in A?

Contrapositives and Monotonic Functions [03/17/2003]
Defining 'monotonic' (increasing or decreasing functions) and 'contrapositive'.

Creating a Mathematical Model of a Complicated Situation [10/09/2005]
I am riding a bike and have several possible routes, each of which contains various traffic lights, amounts of traffic, the possibility of being stopped by a policeman for running a light, and other factors which influence the time it will take to ride each route. How can I model the routes to predict which will be the best choice?

Crossing a Canyon [5/10/1995]
Basically, we're trying to cross a canyon. From a point on one side, a rope stretches across and drops ten feet vertically...

Curvy ... and Topsy-Turvy? [11/22/2014]
Inspecting the zeros of a polynomial, a teen wonders why its plot resembles the graphs of its constituent terms, but with the signs reversed. Doctor Peterson explains the contrary curves with numerical approximation; Doctor Greenie, with a logical extension.

Defining Positive Zero [03/07/2003]
The definite integral of f(x) from 0 to x has exactly one positive zero at x = a. What's a 'positive zero'?

Defining Transcendental Functions [02/05/2001]
Can you give me a clear definition of "transcendental function"? How are algebraic, elementary, and transcendental functions related?

Degree of Constant Function [11/08/2001]
We think F(x) = 1x^0 is not a polynomial function (because polynomials shouldn't have discontinuities), but F(x) = 1 is a polynomial. And F(x) = 1 still has degree 0 but for reasons we can't explain.

Dependent and Independent Variables [05/24/2005]
A look at how functions can be written in different ways, leading to different variables being considered dependent or independent.

Derivative of an Odd Function [02/27/2002]
How do you show that the derivative of an odd function is even?

Derivatives [12/07/1998]
Can you write x^2 = x+x+...+x (x times) if x is not a positive integer?

Derive the Function of Celsius to Fahrenheit [09/22/1997]
How do you get the equation of Celsius to Fahrenheit and vice versa? How would you graph that function?

Determining Roots of a Function [04/03/2003]
I know that under specific conditions we can calculate the roots of a function using the Newton-Raphson Method. The Newton-Raphson method uses the tangent to approximate the root. I want to know if there is a method to approximate the root with a parabola instead of the tangent.

Determining whether a Function is Even or Odd [11/09/1999]
How do you determine whether a function is an even function, an odd function, both, or neither?

Differentiating a Quotient [08/29/1997]
Given that g(x) = x-4/x+3, find: {g (x+h) - g(x}/h).

Differentiating Variables from Functions [02/03/2004]
How can you tell whether a letter stands for a variable or for a function?

Dilation Designation: Why in the Family of Trigonometric Functions Rather than Quadratics? [09/08/2016]
An adult wonders why horizontal dilation falls under trigonometry rather than, say, under quadratic function families: is there something inherent to quadratics that makes such scale factoring redundant? Doctor Peterson algebraically isolates the multiplier, interprets what happens — and reveals the associativity at the heart of the matter.

Dilations of the Graph of y = f(x) [08/22/2004]
If y = 2f(x) stretches y = f(x) vertically by a factor of 2, why does y = f(2x) shrink it horizontally by a factor of 2? The notation seems inconsistent.

Discrete Functions, Broken Down [09/14/2016]
Given written descriptions of everyday situations, a teen struggles to tell the difference between discrete functions and continuous ones. Doctor Peterson guides her to solve a simpler problem.

Discrete or Continuous Problem [2/13/1995]
Mae spends $65 a week on food. After 3 weeks she has $415 left in her food budget. What my math class wants to know is: Is this continuous or discrete?

Discussion of Euclidean Functions of Z [06/10/2008]
Can you help me give a description of all Euclidean functions of Z? The common example is of course the absolute value function, but it seems to me that other weird Euclidean functions can be constructed, too.

Does the Order Matter When Transforming a Function? [08/31/2007]
A function such as -2(x - 5)^3 + 3 has a reflection, a vertical stretch, a horizontal shift, and a vertical shift. How do you decide in which order to perform the translations? Does it matter?

Domain and Natural Domain: What's the Difference? [01/13/2010]
Doctor Rick produces a function with an undefined value in its natural domain -- then declares a more restrictive domain.

Domain and Range [08/25/1997]
Can you please explain about domain and range?

Domain and Range of a Function [03/05/2004]
Given f(x) = 2x, find the domain and range of the function.

Domain and Range of an Inverse Function [03/18/1998]
Why is the domain of a function equal to the range of the inverse function?

Domain and Range of Functions [03/20/2003]
I am learning in school about the domain and range of functions. Why are they called domain and range? And where did they come from?

Domain and Range of x^x [08/24/1998]
What is the domain and range of x^x?

Domain and Range the Inverted Way [05/12/2017]
Typos throw off a student's understanding of the domain and range of a function with an unknown under a radical in the denominator of a fraction. To confirm what graphing suggests about the range of the original function, Doctor Peterson resorts to calculating the inverse, examining its domain, and considering the introduction of extraneous solutions.

Domain, Asymptotes, Intercepts of a Function [04/01/2003]
What is the domain of this function? What asymptotes does it have? What are the x and y intercepts? Etc...

Domain of a Composite Function [08/30/2001]
I don't completely understand how to find the domain of a composite function. Could you please explain it?

Domain, Range, and Asymptote [10/12/1998]
How do you find the domain, range, and asymptote of a function?

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