For the function f : [1, ) [1, ) defined by f(x)=(x-1)4+1, among the two statements : (I) The set S=\x [1, ) : f(x)=f-1(x)\ contains exactly two elements, and (II) The set S=\x [1, ) : f(x)=f-1(x+1)\ is an empty set,
- only (I) is TRUE
- only (II) is TRUE
- both (I) and (II) are TRUE
- neither (I) nor (II) is TRUE
Answer
(1) only (I) is TRUE
Solution
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f(x)=(x-1)4+1 is strictly increasing on [1, ) and onto [1, ), so f-1 exists:
y=(x-1)4+1 f-1(x)=1+(x-1)1/4
Statement (I):
f is strictly increasing, f(x)=f-1(x) f(x)=x
(x-1)4+1=x (x-1)4=(x-1)
(x-1) [(x-1)3-1 ]=0 x=1 or x=2
S =\1, 2\ has exactly two elements; (I) is TRUE.
Statement (II):
f-1(x+1)=1+x1/4, so f(x)=f-1(x+1) (x-1)4=x1/4
Let h(x)=(x-1)4-x1/4, continuous on [1, ).
h(2)=1-21/4<0 and h(3)=16-31/4>0
h(x)=0 for some x (2, 3) (intermediate value property)
S is not empty; (II) is FALSE.
Hence only (I) is TRUE.
Answer: (1)