Function question

neelgandham: Community Manager; Posts: 1060; Joined: Fri May 13, 2011 6:46 am; Location: Utrecht, The Netherlands; Thanked: 318 times; Followed by:52 members

by neelgandham » Thu Nov 17, 2011 9:17 am

Though not really sure if Inverse of a Function is a topic which is tested in GMAT, here is how I think we should approach!

f(x) = ax + b

Assumption: f-1(x) = Inverse of a function f(x)

x = (f(x)-b)/a, now replace x with f-1(x) and f(x) with x (definition of inverse of a function)
f-1(x) = (x-b)/a

f(1) = 1 => a (1) + b = 1 => a + b = 1
f-1(5) = (5-b)/a = 2 => 2a + b = 5
from the above equations, a = 4 and b = -3

f(x) = 4x - 3
f-1(x) = (x+3)/4

f-1(7) = (7+3)/4 = 5/2
f-1(-11/2) = ((-11/2)+3)/4 = -5/8

Anil Gandham
Welcome to BEATtheGMAT | Photography | Getting Started | BTG Community rules | MBA Watch
Check out GMAT Prep Now's online course at https://www.gmatprepnow.com/

Quote

GmatMathPro: GMAT Instructor; Posts: 349; Joined: Wed Sep 28, 2011 3:38 pm; Location: Austin, TX; Thanked: 236 times; Followed by:54 members; GMAT Score:770

by GmatMathPro » Thu Nov 17, 2011 9:32 am

f(x)=ax+b
f(1)=1, so 1=a(1)+b or
a+b=1

f^-1(5)=2

This says that if we plug 5 into the inverse function we get 2. the inverse function essentially switches all of the x and y values of the original function. So, for example, if the original function contains the point x=4, y=10, then the inverse function contains the point x=10, y=4. f^-1(5)=2 tells us that the inverse function contains the point x=5, y=2. Therefore the original function contains the point x=2, y=5. Plug this in to the original function to get 5=2a+b. Now we have

2a+b=5
a+b=1

solve this system to get a=4, b=-3. So f(x)=4x-3.

To find f^-1(x), which is the inverse function you essentially need to switch x and y in the equation and solve for y:

f(x)=4x-3
y=4x-3
x=4y-3 (switching the x and y)
y=(x+3)/4 (solving for y)
f^-1(x)=(x+3)/4 (switching back to function notation)

Now, we can find the last two values by plugging in:

f^-1(7)=5/2

f^-1(-5.5)=-5/8

For any bystanders, I should note that I don't think inverse function problems like this are tested on the GMAT

Pete Ackley
GMAT Math Pro
Free Online Tutoring Trial