We are often given a description of a functional relationship between
different variables and have to figure out what it means. This
involves making the conversion between variables and the letters in
the expression being described, constants and other letters, and
determining how the relationship would be represented graphically.
When an expression contains a combination of letters and numbers some
of the letters may represent constants and some may represent
variables. For example, in the phrase
f(x) = a sin(b x),
the a and b are thought of as constants, the x is
the variable, and f represents the functional operation
itself. Often the letters at the beginning of the alphabet,
a,b,c... stand for constants, those in the middle,
f,g,h... for functions, and those at the end, ...w,x,y,z
for variables. However, in each specific case you will either be told
which is which or be expected to decide from context.
Examples
- The depth, y (in meters), of water in a cove as
a function of time t (in hours since midnight) is given
by
where y0 is the average
depth in the cove. Here the variables are t (the
independent variable) and y (the dependent variables) and
all of the other letters are constants. It is common for
subscripts to denote a specific (fixed) value of the variable.
- The equation for a line through a fixed point
(h,k) with a specified slope, m, is given
by
In this case the variables are x (independent) and
y (dependent). The slope and coordinates of the point are
treated as constants.
The word coefficient describes a constant which multiplies a
variable.
Example
- In the formula for the volume of a sphere,
the coefficent of 
is the constant 
.
Many problems begin by asking us to set up an equation which describes
a relationship between two quantities. Some of these relationships
are well known formulas, like the one relating the volume of a sphere
to its radius (given in the example above). The formula above is a
convenient form for finding V from r: r is the
independent variable and V is the dependent variables. We say
that the formula gives V as a function of r,
or, that it gives V in terms of r. We could
equally well have chosen to write an equation for the radius as a
function of the volume, r(V), in which would mean having
V as the independent variable and writing
![r = [(3/(4pi)) V]^(1/3)](images/9.1.e.4.gif)
.
Examples
- After a liquid fertilizer is applied to a house plant, the
concentration of fertilizer in the plant is found to be twice
the reciprocal of the amount of time elapsed since the
application. Express the concentration, C, as a function
of time, t.
We want
- A bus rental company charges a $150 initial fee for a bus. In
addition to this, the company charges $23.00 an hour for
operating costs and $10.00 an hour for the driver's wages.
Write a function D(h) for the total cost in
dollars, D, in terms of the rental time in hours,
h.
- The volume of a right circular cone is given by
Give a formula for the volume as a function of the radius for a
cone whose height is double its radius.
We know that height = (twice)(radius), so h =
2 r. Substituting into the volume formula gives
In addition to setting up a function, we often want to sketch its
graph. We customarily use the horizontal axis for the independent
variable and the vertical axis for the dependent variable.
Example
- Label the axes to sketch a graph of the temperature T
(in degrees) of a glass of water t minutes after it is
placed in the refrigerator.
Here the independent variable is t and the dependent
variable is T, so the axes would be labeled:
precal: 9.1 - setting up functions
page created: Fri Nov 22 19:43:57 2024
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