Expert: Paul Klarreich - 11/16/2005

Question
Studying College Algebra

Suppose that j(x)= h^-1(x) and that both j and h are defined for all values of x.  Suppose also that h(4)=2 and j(5)=-3.  Evaluate the following expressions if possible.  (The h above is the inverse function. I wasn't sure how to do that on the computer)
(a) j(4)  (b) h(j(4))  (c) j(2)  (d) h^-1(-3)
(e) j^-1(-3) (f) h(5) (g) (h(-3))^-1
(h) (h(2))^-1  
So, if you have anything to tell me how to understand inverse functions better so this makes sense, I appreciate it.  Thank you very much for your time.

Answer
Hi, Colleen,

Suppose that j(x)= h^-1(x) and that both j and h are defined for all values of x. Suppose also that h(4)=2 and j(5)=-3. Evaluate the following expressions if possible. (The h above is the inverse function. I wasn't sure how to do that on the computer
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Close enough.  I know what you mean.  Here are the basic ideas of inverse functions:

1. If  f  is a function, then it has a set of inputs (the domain) and a set of outputs (the range).
2. If the inverse of f (yes, we can write f^-1) is a function (and this isn't always so) then it also has a domain and range.  The domain of f^-1 becomes the range of f, and vice versa.
3. So the two functions just switch their inputs and outputs.
4. So f^-1(x) would have the property that  f^-1(f(x)) = x and f(f^-1(x)) = x, for all x, providing these things are defined.  In other words, if you supply an input to one of them, then take the output and supply it to the other, you get back your original number.

In your example above, where j is the inverse of h,
h(4) = 2  would imply that j(2) = 4
j(5) = -3 would imply that h(-3) = 5

So we can try the following:


(a) j(4)  not defined.
(b) h(j(4)) = 4, as described above.
(c) j(2) = 4 (likewise)
(d) h^-1(-3) = j(-3) but you don't have any info on this
(e) j^-1(-3) = h(-3) = 5
(f) h(5) not defined.

(g) (h(-3))^-1 = 1/h(-3) = 1/5.  Now this use of "^-1" is different in meaning.  If you put the "-1" exponent RIGHT AFTER the function name, you mean the inverse.  If you put it after a number, it means the reciprocal of the number.  And, while 'h' is the name of a function, h(x) is a number.

(h) (h(2))^-1  likewise, but h(2) is not defined.