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Composition of Functions


    To compose means to create. A composition is what is created. Very often a piece of music or work of art is called a composition.

    In math, functions are very often compositions.

    How would one compose a function? Why would one compose a function?

    Consider the why first.

Why Compose A Function?
  • Because a function is needed to do a job.
  • Because a function is needed to describe and predict behavior -- a more sophisticated job.
  • For fun or to create something of beauty.

Because a function is needed to do a job.

    These jobs might be completed by anyone or unique to the composer. The most well known and useful functions are the operations -- addition, subtraction, multiplication, division, square root, absolute value (absolute value is a composition). Next most useful are the compositions which are coded by a formula -- area, perimeter, volume, distance, and so on.

    Please visit the Math Functions Page now so you will understand the vocabulary and examples explained below. You might wish to keep the page open for reference.

Some Functions that Use One Value Some Functions Which Require Two Values Some Functions Which Require More than Two Values

    The name of the function identifies the kind of job the function peforms. Some functions are used by the general public and some are used by those in mathematics, science, and engineering.

    The functions above have been sorted by the number of arguments required before the function can be evaluated and the work on the specific values completed.

    For example, the notation P(L, W) = 2(L) + 2(W), says, to compute the perimeter (of a rectangle), take twice the length and twice the width and add these values. The arguments are the length and width. It took two multiplications and an addition to complete the function. This function, as many others, is built or composed using other functions, here, the multiplications and addition.

    The notation V(l,w, h)= lwh, says, to compute the volume of a rectangular box multiply the length, the width, and the height.

    Normally these functions have constants as arguments. Sometimes, however, functions have other functions as arguments. More on that below.

Because a function is needed to describe and predict behavior.

Ex. 1

    Above, V(l,w, h)= lwh said multiply the length, the width, and the height to get the volume. We'll use this function to compose a new one to answer the question below.

    Build an open box from a 24 by 24 square piece of paper. What's the maximum volume?

    V(l,w, h)= lwh becomes v(x)=(24-2x)(24-2x)(x) where x is the height of the box and 24 -2x is both the length and the width. A function has been composed to solve a specific problem.

    V(x)=(24-2x)(24-2x)(x) expanded is 4x3 - 96x2 + 576x. Settting the derivative of this equal to 0 and solving, x is 12 or 4. Since a height of 12 would not make a realistic box, the height of the box should be 4. That makes the maximal dimensions 16, 16, and 4 units and the maximum volume 1024 cubic units.

Because a function is needed to describe and predict behavior.

Ex. 2

    The formula s(t) = (a/2)t2 + v0t + s0 is a function which models projection -- shooting, propelling, dropping stuff at a certain speed, from a certain height. The model does not include wind resistance and the initial trajectory of the object.

    The symbol s is used for the height or displacement. The symbol t is used for time. The symbol s(t) says height as a funtion of time.

    The function (a/2)t2 + v0t + s0 is the composition, addition, of a constant, a linear, and a quadratic function. To describe a specific situation, values for a, v0 (the initial velocity), and s0 (the initial height) are used to tailor the formula to the specific situation.

s(t) = -16t2 + 0t + 60

    This functions expresses the behavior of an object dropped on Earth from a height of 60 feet above.

    For more information on how quadratic functions are used to model motion, see The Quadratic As A Model For Projections.

For fun or to create something of beauty.

    The sine function is valuable tool in mathematics and engineering. Its graph is beautiful and an ideal function to use to compose other playful and beautiful functions. Below are two examples.

    In the first composition, 1.5sin(.2x²), the augument of the sine is a flattened parabola which gives the graph symmetry and decreases the "speed" at which the sine goes through its cycle. The multiplication of this by the constant 1.5 gives the function a bit more height compared to the height of the sine function.

    In this second compostion, the height will continue to grow because the natural log function, seen as ln(x+4), continues to grow and here multiplies the height of the sine.

    The curve is shifted to the left by the linear function x+4 seen as the arguments of the natural log and the sine.


    Remember: How would one compose a function? Why would one compose a function?

    Now consider how.

    One may add, subtract, multiply, divide, or take functions of functions. The only thing to really watch is restrictions on the functions used and the functions composed.

    Stated in function notation, one may use two functions and take the:
sum (f + g)(x) -- add the functions
difference (f - g)(x) -- subtract functions
product (f · g)(x) -- multiply functions
quotient (f ÷ g)(x) -- divide functions
composition (f o g)(x)= f(g(x)) -- take a function of a function
    Speaking of the graph of a function as a refection of the composition's behavior:
add the functions -- shift the function to right or left or up or down
subtract functions -- shift the function to right or left or up or down
multiply functions -- zeros remain, shape is "more wiggly or flat"
divide functions -- zeros remain or asymptotes are created
take a function of a function -- one function is the domain of the other.

    Examine the examples.

f(x) g(x) Composition Description
x 2 x + 2, f(x)+g(x) shifts x "up 2"
x 2 x-2, f(x)-g(x) shifts x "down 2", See plot
x x x², f(x)·g(x) parabola, zero at 0, concave up, square of a number, x²
-1 x -x, f(x)·g(x) "flips" x, zero at 0, opposite of a number, -x
x+2 (x+2)²,
first add 2
then square
parabola, zero at -2, "shifts left 2"
x+2 x²+2;
first square
then add 2
parabola, zero at 0, "shifts up 2"
x+1 x (x+1)/x,
rational function, zero at -1,
vertical asymptote at x=0, horizontal asymptote at y = 1,
x may never equal 0, see analysis & graph
x (x²), g(f(x)) absolute value, |x|, square root of the square, by definition by composition

    For more on composition of functions see the links below.

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