Question

Help with graphing problem? will award medal for best answer!

Graphs of irrational functions show things about extraneous solutions. Let f(x) = x - √x-2

a. find f(11), f(6), f(2), f(2), and f(0).
b. plot the graph of the function.
c. what is the domain of x if f(x) is to be a real number?
d. show that there are no real values of x for which f(x)=0.

please help! If you could plot the graph and answer all that would be awesome! Im so confused

Answer 1

Let f(x) = x - √x-2

a. to find each of these answers, you need to "plug and chug" each value into the function. For example f(11) = 11 - √11-2 = 11 - √9 = 11 - 3 = __8__
f(6) = 6 - √6-2 = 6 - √4 = 6 - 2 = ______
Just do the same with the rest of them. Wherever the x is in the function, replace it with the value given in the parentheses.

b. plot the graph of the function.
An easy way to plot a graph, is to use find data points or use data points you already know and then basically "connect-the dots". Using your answers from part (a) is a great way to do this. For example, with f(11), we were able to find that when x = 11, the function (or y) = 8. Therefore, one point in your coordinate graph will be (x, y) = (11, 8). Use the rest of your answers from part (a) to finish.

c. what is the domain of x if f(x) is to be a real number?
The key to answering this part is recognizing and knowing the definition of a "real number". A real number is defined as including all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers such as √2. Basically it encompasses all types of numbers. ) http://www.scimathmn.org/stemtc/sites/default/files/images/frameworks/math/8.1.1A/image131.jpg). Secondly, it is important to know the definition of domain- this refers to all the possible numbers that x could be. Lastly, one other important rule to keep in mind, is this: you may not have a negative number under a square root (when this happens it is known as an imaginary number). Therefore, x may not be less than 2. Hence, x must be between (the domain is) +2 and infinity ("greater than or equal to 2").

d. show that there are no real values of x for which f(x)=0.
This is a simple proof done by assuming that there is in fact a value for x for which f(x)=0. Therefore, given the function f(x) = x - √x-2 you set up the equation as:
0= x - √x-2 and solve for x. You will find that it does not work and will end up with an inequality/false statement. After arriving to a false statement you could state. "QED" or "therefore there are no real values of x for which f(x)=0.