http://www.ck12.org Chapter 6. Transcendental Functions
It is important to note that for the functionf(x) =x^2 to have an inverse, we must restrict its domain to 0≤x<∞,
since that is the domain in which the function is increasing.
Continuity and Differentiability of Inverse Functions
Since the graph of a one-to-one function and its inverse are reflections of one another about the liney=x,it would
be safe to say that if the function fhas no breaks (no discontinuities) thenf−^1 will not have breaks either. This
implies that iffis continuous on the domainD,then its inversef−^1 is continuous on the rangeRoff.For example,
iff(x) =√x, then its domain isx≥0 and its range isy≥ 0 .This means that f(x)is continuous for allx≥ 0.
The inverse off(x)isf−^1 (x) =x^2 ,where its domain is allx>0 and its range isy≥ 0 .We conclude that iffis a
function with domainDand rangeRand it is continuous and one-to-one onD,then its inversef−^1 is continuous
and one-to-one on the rangeRoff.
Suppose thatfhas a domainDand a rangeR.If fis differentiable and one-to-one onD,then its inversef−^1 is
differentiable at any valuexinRfor whichf′(f−^1 (x)) 6 =0 and
d
dx[f− (^1) (x)] =^1
f′(f−^1 (x)).
The formula above can be written in a form that is easier to remember:
dy
dx=
1
dx/dy.In addition, iff on its domain is either f′(x)>0 orf′(x)< 0 ,then fhas an inverse function f−^1 andf−^1 is
differentiable at all values ofxin the range off.In this case,f−^1 is given by the formula above. The example below
illustrate this important theorem.
Example 4:
In Example 3, we were given the polynomial functionf(x) = 3 x^5 + 2 x+1 and we showed that it is invertable. Show
that it is differentiable and find the derivative of its inverse.