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Function question
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- neelgandham
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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
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(x) = ax + b
x = (f(x)-b)/a, now replace x with f-1(x) and f(x) with x (definition of inverse of a function)Assumption: f-1(x) = Inverse of a function f(x)
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
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- GmatMathPro
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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
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