For each of the following functions f: R rightarrow R and each set T R, find f(T

Question

For each of the following functions f: R rightarrow R and each set T R, find f(T), f^-1 (T), f(f^-1 (T)) and f^-1 (f(T)). (1) Let f: R rightarrow R be defined by f(x) = (x + 1)^2 for all x elementof R, and let T = [-1,1]. (2) Let f: R rightarrow R be defined by f(x) = (x + 1)^2 for all x elementof R, and let T = [-5,2]. (3) Let f: R rightarrow R be defined by f(x) = |x| for all x elementsof R, where |x| is the smallest integer greater than or equal to x, and let T (1,3). (4) Let f: R rightarrow R be defined by f(x) = x for all x elementsof R, where x is the greatest integer less than or equal to x, and let T = [0,2] (5,7).

Explanation / Answer

1) f(T)=[0 , 4]

2) f(T)=[0 , 16]

3) f(T)={2 , 3}

4) f(T)={0 ,1 , 2 , 5 , 6}

Explanition:

Here in ex 1 and 2 f(x) is continuous. Moreover we can find the value at which f(x) is minimum and it is -1.

Minimum value is zero. Maximum values are 4 and 16 respectively.

Answer for ex 4 and 5 can be obtained by considering the graph of given function(Step function).

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