Emergency Nursing, Essay Example

• What is a function?

A function is defined as a relation between a given set of elements known as the domain and another set of elements known as the codomain. The function associates each element in the domain with another element in the codomain. The elements in the relationship may be of any kind of thing including objects, words and qualities but are typical mathematical quantities, like real numbers.

An example f(x) =3x is a function with the real numbers as  its domain and codomain because it  associates every real number with the real number thrice  as big and therefore it can be written as  f(2) = 6   Ruthing, D. (1984).

• What is a linear function?

A linear function is a function that has no exponents other than one and is without products of the variables for instance y=x+2, 2x-4y = 1/4 and y= -2, are all linear.

These functions have x as the input variable, and x is raised only to the first power.

Such functions yield graphs that are straight lines therefore, the name linear.

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• What form does a linear function take? (I.e.,What is the standard mathematical notation of a linear function?)

There are three standard forms for linear functions:

• The “general form” Ax + By = C (the equation defines y implicitly as a function of x as long as B? 0.)
• The “Taylor” or “point-slope” form y – yo = m(x – x0) or, equivalently, f(x) = y0 + m(x – x0) (),
• and  The “slope-intercept” form y = f(x): f(x) = mx + b (),

In the general form, -A/B is the slope of the line   if B? 0 and infinite if B = 0.

If f(x) is linear then the graph of y = f(x) is a straight line. The parameter m in the two formulas is the slope of this line. In the point-slope form, the point (x0, y0) is a point on the line y = f(x). In the slope-intercept form, the parameter b is the y-intercept. (oregonstate.edu/instruct/mth251/cq/…/linear/lesson.html)

• What is the formula for determining the slope of a line?

The slope of a line is the ratio of the change in y over the change in x. the formula for finding the slope of a line is therefore as follows

m=y₂-y1?x₂-x₁

• Suppose you have a lemonade stand, and when you charge $1 per cup of lemonade you sell 60 cups. But when you raise your price to $1.50 you only sell 30 cups. Write an equation for the number of cups you sell as a function of the price you charge. Denote “C” for number of cups, and “P” for the price you charge. Assume the function is linear.

Solution:

Let the function be C (P) = Px + y, where x and y are constants to be determined.

When P = $1, C = 60 60 = x + y …………(i)

When P = $1.5, C = 30 30 = 1.5x + y ………(ii)

Solving the equations by Subtracting (i) from (ii) we get the value of x as   = -60

By replacing the value of x   in equation (1), we get y =120

Thus, the function is C (P) = -60P + 120 = 60(2-P)

• x -3 -2 1 3 4

f(x) 0 3 12 18 21

y = f(x) = mx + b

Take two sets of the corresponding values from the table e.g

When f(x) =o, x=-3

Therefore 0=-3m+b…….eqn1

and another set in which f(x) =21 and x=4

The equation will be 21=4m +b……eqn.2

Solving the two equations by subtracting equation 2 from equation 1 gives m=3

Replace the value of m to either equation and you get the value of m as m=9

Therefore the equation for f(x) is f(x)=3x+9

• Equation a. is a function because it has real numbers as  its domain and codomain because it  associates every real number with the real number eight times   as big and therefore it can be  written as  f(2) = 16

b. f(x) = 45 if x>2 otherwise f(x) = -4 is not a function because for the first part there is a range of values greater than two whose values are different and therefore give different answers after computation and so is the case with the values which are less than two. According to the definition of a function one real number in the domain should be associate with another one in the codomain.

f(x) = 4 if x>0 or f(x) = -4 if x<0 or f(x) = 4 or -4 if x = 0 the last part of the equation f(x) = 4 or -4 if x = 0 is a function which can be written as f(x)=x + the square root of 16 since the value of x is zero and the square root of 16 is ether 4 or-4 but the first two parts do not have specific values in the domain association with specific ones in the codomain since values below and above 4 are varied. (oregonstate.edu/instruct/mth251/cq/…/linear/lesson.html)

• For each of the relationships below, explain whether you think it is best described by a linear function or a non-linear function. Explain your reasoning.

How fast it takes you to get to work as a function of how fast you drive is a linear function, because  time = distance / velocity. Since the distance to the work place is fixed, the function may be written as t(v) = d/v, with v = velocity, t=time and d = distance and therefore the higher the speed, less the time it takes to cover the distance which is fixed and faster you get to your work.

• Probability of getting into a car accident as a function of how fast you drive.

This is a non linear function, since there are many other factors which may lead to occurrence of accidents like the state of the vehicle among others r the speed, higher is the probability of getting into an accident.

• Your height as a function of age (from age 0 to 100)

This is not a linear function because rate of growth has phases .in the teenage age its called the log phase where growth is very fast and slows as one ages. Teenagers experience a rapid increase in height with the rate slowing and even coming to a halt with age. 

References

Ruthing, D. (1984), “Some definitions of the concept of function from Bernoulli, Joh. to Bourbaki, N.”, Mathematical Intelligencer 6 (4): 72–77. web. Retrieved  March 3, 2010 Available: <id.mind.net/…/functionInstitute/linearFunctions/linearFunctions.html>

Web. Retrieved March 5, 2010. Availble: <oregonstate.edu/instruct/mth251/cq/…/linear/lesson.html)>