Writing Functions

Writing Functions

_ as a function of _
The first blank represents the dependent variable, and the second blank represents the independent variable. We are used to seeing y as a function of x.

This could also be represented as



The two blanks could be any sort of relationship.

Example:
Write the area of a circle as a function of its radius. There are usually two equations needed to complete this type of problem. In this case, the standard equation is already in the terms of what is asked, so only one equation is needed.

This equation would be the area of a circle as a function of its radius: the radius is the independent variable, and the area is the dependent variable, as is the requirement for this type of the problem.




Example:
Write the area of an equilateral triangle as a function of the height of the triangle. There are two equations needed for this problem. The first one is the equation for the area of a triangle:



The next equation needed is the one that will put the base in terms of height. First, you must bisect the triangle.

You will end up with two 30-60-90 triangles with the adjacent sides of x and h. The hypotenuse would be one of the original sides of the triangle, s. One of the adjacent sides (x) would be 1/2s because the triangle was bisected. So,



According to the laws of a 30-60-90 triangle, if the side across from the 30 degree angle is called x, then the side across from the 60 degree angle is
. The side across from the 60 degree angle is h in this triangle, so




So you can substitute




in the above equation for h to get:



Once you solve the equation for s, you get



Recall the equation for the area of a triangle. Since the base is s, the equation would be



The equation is supposed to be in terms of only h, so you substitute to get:




Then you simplify to end up with the final equation




Which is the area of an equilateral triangle as a function of its height.


Works Cited:
"CONCEPT OF C PROGRAMMING: C - Area and Circumference of Circle." CONCEPT OF C PROGRAMMING: C - Area and Circumference of Circle. N.p., n.d. Web. 14 Dec. 2014.
 "Project Euler 94: Almost Equilateral Triangles with Integral Sides and Area." Mathblog.dk. N.p., 28 Apr. 2012. Web. 14 Dec. 2014.