Restricted Domains (2 Viewers)

[king_of_boredom]

ok BOS mathematicians, need your help this time.

i just started inverse functions last lesson, and have a 'lil trouble grasping the concept of 'restricted domains' (that is, the domain of a function on order for it's inverse to be a function). how do you figure it out if given a question? i figured a way out, dunno if it's right, but what you do is

a) find the inverse function
b) find the range of the inverse function
c) change the y to x, hence this gives your 'restricted domain'.

for example:

Q: state the restricted domain of y = x^2 + 1.

A:

a) ok finding inverse function...
f-1(x): x = y^2 + 1
y^2 = x - 1
y = sqrt(x - 1)

b) ok, so range of inverse function is y (is greater than or equal to) 0

c) therefore restricted domain is x (is greater than or equal to) 0

am i doing this right?

thx in adv

 

[acmilan]

Easiest way to do it is graph the function. Then choose a domain where for every y value there is at most 1 x value. This can be tested by drawing horizontal lines and making sure the line only cuts the graph at most one time in the chosen domain. There could be more than one domain to choose from, or in cases such as trig functions, infinite number of domains to choose. I dont think your method will necessarily work in all cases.

If you take your example of y = x^2 + 1,

graph it and you'll see the a horizontal line anywhere on the graph will only cut the graph once for x > 0 and for x < 0

 

[king_of_boredom]

If you take your example of y = x^2 + 1,

graph it and you'll see the a horizontal line anywhere on the graph will only cut the graph once for x > 0 and for x < 0

so does that mean that there are 2 restricted domains? i think the terms 'monotonic increasing' and 'monotonic decreasing' ring a bell for me here...lol wat do they mean again?

 

[acmilan]

If a function is always increasing or always decreasing on a domain then it will always have an inverse.

One way to think of it is that it can never 'turn back' if it is always increasing or always decreasing ie. f'(x) > 0 always. Since it cant turn back, there will always only be at most one x value for each y value

 

[king_of_boredom]

ah, i see. so would you suggest that i graph the function whenever i get a question of this nature?

 

[acmilan]

ah, i see. so would you suggest that i graph the function whenever i get a question of this nature?

It would probably be the best method to visualise the domain

 

[king_of_boredom]

ah ok thanks dude, i think i get it now. yeah graphing and using the horizontal line test would be best, especially with inverse trig functions which i have now started.






p.s 100th post! yay