http://www.ck12.org Chapter 6. Transcendental Functions
The Inverse of a Function
We discussed above the condition for a one-to-one function: for each output, there is only one input. A one-to-one
function can be reversed in such a way that the input of the function becomes the output and the output becomes an
input. This reverse of the original function is called theinverseof the function. Iff−^1 is an inverse of a functionf,
thenf−^1 ◦f=x.For example, the two functionsf(x) = 2 x+3 andh(x) =x− 23 are inverses of each other since
f◦h=f(h(x)) = 2[x− 3
2]
+ 3 =x− 3 + 3 =x,h◦f=h(f(x)) =(^2 x+ 23 )−^3 =^2 xx=x.Thus
f◦h=h◦f=x,andfandhare inverses of each other.
Note: In general,f−^16 =^1 f.
When is a function invertable?
It is interesting to note that if a functionf(x)is always increasing or always decreasing over its domain, then a
horizontal line will cut through this graph at one point only. Thenfin this case is a one-to-one function and thus
has an inverse. So if we can find a way to prove that a function is constantly increasing or decreasing, then it is
invertableormonotonic. From previous chapters, you have learned that iff′(x)>0 thenfmust be increasing and
iff′(x)<0 thenfmust be decreasing.
To summarize, a function has an inverse if it is one-to-one in its domain or if its derivative is eitherf′(x)>0 or
f′(x)< 0.
Example 2:
Given the polynomial functionf(x) = 3 x^5 + 2 x+ 1 ,show that it is invertable (has an inverse).
Solution:
Taking the derivative, we find thatf′(x) = 15 x^4 + 2 >0 for allx.We conclude thatf(x)is one-to-one and invertable.
Keep in mind that it may not be easy to find the inverse off(x) = 3 x^5 + 2 x+1 (try it!), but we still know that it is
indeed invertable.