Ask Dr. Math

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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Graphing f(2x) and f(|x|) [09/03/2003]

Given f(x), how do you graph f(2x) and f(|x|)?

Graphing Limits [09/12/2001]

I was hoping you could explain the concept of "limits" and how to read them from graphs.

Graphing Piecewise Functions [09/09/2001]

I do not understand how to graph piecewise functions.

Graphing Polynomial Functions in Factored Form [07/07/2004]

When graphing polynomial functions, why do factors of even multiplicity cause the graph to not cross the x-axis, while factors of odd multiplicity cause it to cross?

Graphing Rational Functions and Vertical Asymptotes [03/28/2008]

When working with rational functions such as y = (x-5)/(x-3), how do you know if the graph curves up or down at the vertical asymptote?

Graphing Reciprocal Functions [11/01/2001]

I know what a reciprocal is, but I don't know how to graph it.

Graphing Sin and Cosine Functions [5/24/1996]

I really need some help in finding the graphs of the following functions...

Graphing the Absolute Value/Square Root of a Function [07/26/2002]

Can you illustrate and discuss how taking the absolute value or the square root of a function affects the graph of the function?

Graphing Two Functions With One Equation [10/21/2003]

I have a question on graphing functions. I am just wondering if it is possible to use one equation to graph two functions?

Graphing Two Functions with One Equation? [10/21/2003]

Suppose I have two functions, like y = sqrt(x) - 4 and y = sqrt(x) - 8. Is there a single equation that would produce the same graph as the two functions graphed together?

Graph of Circle as a Function [05/30/1998]

The graph of a circle is not a function because it fails the vertical line test. How could you make it a function?

Graphs of Inverse Functions [07/25/2002]

Explain why, if y=f(x) and y=g(x) are inverses, the graph of either function is the graph of the other reflected across the line y=x.

Graphs of Step Functions [11/29/2003]

Why do some graphs move in steps rather than smooth lines? What causes that?

Graph with f(x) [10/17/2001]

Sketch the equations: y = f(x)+ 2; y = f(x-3); y = 2f(x).

Greatest Integer Equation [08/06/2003]

I am trying to correctly interpret [[x]]^2 + [[y]]^2 = 1, where f(x)= [[x]], is the Greatest Integer function.

Greatest Integer Function [12/04/2002]

How can I explain greatest integer function to my home-schooled son?

Greatest Integer Functions [09/27/1998]

Can you help me solve for the graph of [y]=[x], where [] is the greatest integer function?

The Heaviside Step Function [09/03/2003]

How can I construct a term that is zero when the input is below some limit, but nonzero above it?

How Are Functions and Expressions Related? [07/09/2004]

What is the relationship between a function and an expression? I don't see any relationship, they are two completely different things.

How to Evaluate 4!!!! [10/16/2003]

Our Calculus 2 teacher has challenged us to find the exact value for 4!!!!, but the number is too large to reasonably calculate by hand.

How to Find the Range of a Function [02/25/1998]

How do you find the range of a function like g(x) = (x+1)/(x^2-1)?

How to Solve Equations with No Analytic Solution Method [10/26/2005]

A discussion of solving equations that can't be solved analytically by using iterative estimation methods including bisection, false position, and Newton's method. The equation x(e^x) = 3 is used as an example.

Hyperbolic Functions [06/27/1998]

What does the inverse of sinh or cosh mean?

Hyperbolic Functions [10/14/1998]

Can you explain hyperbolic functions? How are they defined? How do you evaluate them?

Implicit Functions [11/26/1997]

Please give me a definition and several examples of an implicit function.

Importance of Linear Functions [02/06/2002]

What is the importance of linear functions in the real world, and how are they used?

Infinity as a Skolem Function [10/28/2000]

Is infinity an absolute concept, a relative concept, or both?

Inner Product and L-2 Distance [11/14/2001]

Why is "distance between two functions" calculated by multiplying f(x) and g(x) and then integrating with respect to x within the defined domain?

Integer Iteration Function [12/24/2003]

Let X be a positive integer, A be the number of even digits in that integer, B be the number of odd digits and C be the number of total digits. We create the new integer ABC and then we apply that process repeatedly. We will eventually get the number 123! How can we prove that?

Integrating X^x, Closed Form [10/31/1996]

How do you express the equation y = xcosx in terms of y?

Interesting Continuity Question [09/01/2005]

Is every function continuous at some point in its domain?

Intermediate Value Theorem [09/19/2002]

How can we prove by the intermediate value theorem that there is a point on the path that a hiker will cross at exactly the same time of the day hiking up and returning?

Interpolation [03/06/2003]

I cannot remember the formula for interpolating between two numbers.

Inventing an Operation to Solve x^x = y [02/19/2000]

Can we solve problems like x^x by inventing an operation higher than exponentiation? Can the "center" of an operation be defined?

Inverse Functions: Arcsec(x) [02/11/1999]

If y = arcsec(x) then what does x equal? How is this problem related to inverse functions?

Inverse Functions in Real Life [01/25/2002]

I would like to know if you have a good example of how inverse functions would be used in real life.

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.

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