If f(x)=(8-3x)/18, find f^-1(x)

Question

I don't remember how to work problems with negative exponents

Answer 1

In this problem I think that f^-1(x) represents the inverse function, not a negative exponent.  

 

To find the inverse function you have to solve the original equation for x in terms of y.  Remember that f(x) is just another term for y, so the original equation is  

 

y = (8 - 3x) / 18  ==>  

18y = 8 - 3x        ==>  

18 y - 8 = -3x      ==>  

x = -6y + (8/3)   is the inverse function  

 

which can also be written as f(y) = -6y + (8/3) since in this case f(y) is another term for x  

 

We usually write functions in terms of x, not y, so this would be written as  

 

f^-1(x) = -6x + (8/3)  

 

You can check that this is the inverse function by putting a value in for x in f(x), seeing what value it gets mapped to, and then seeing what value the inverse function maps the value you got back into - it should be the same value you started with because that's the definition of what the inverse function does.  For example  

 

f(0) = (8 - 3*0) / 18 = 8/18 = 4/9, so f maps 0 to 4/9  

 

f^-1(4/9) = -6*(4/9) + (8/3) = (-24/9) + (24/9) = 0, so f^-1(x) maps 4/9 back to 0  

 

Now, this single point check by itself doesn't actually prove that these 2 functions are inverses (you would need to show that f(f^-1(x)) = x for all x, which is a topic for another day), but it does give you a good sense that the answer is correct.