#### Question

Let *A* = {−1, 0, 1, 2}, *B* = {−4, −2, 0, 2} and *f*, *g*: *A* → *B* be functions defined by *f*(*x*) = *x*^{2} − *x*, *x* ∈ A and g(x) = `2|x - 1/2|- 1, x in A`. Are *f* and *g* equal?

Justify your answer. (Hint: One may note that two function *f*: *A* → *B* and g: *A* → B such that *f*(*a*) = g(*a*) &mn For E;*a* ∈*A*, are called equal functions).

#### Solution

It is given that *A* = {−1, 0, 1, 2}, *B* = {−4, −2, 0, 2}.

Also, it is given that* f*, *g*: *A* → *B* are defined by *f*(*x*) = *x*^{2} − *x*, *x* ∈ *A* and `g(x) = 2|x - 1/2| - 1, x in A`.

It is observed that:

`f(-1) = (1^2) - (-1) = 1+1 = 2`

`g(-1) = 2|(-1)-1/2| - 1= 2(3/2) - 1= 3 -1 =2`

=> f(-1) = g(-1)

f(0) = (0)^2 - 0 = 0

`g(0) = 2|0 - 1/2| - 1 = 2(1/2) - 1= 1 - 1 = 0`

=> f(0) = g(0)

f(1) = (1)^2 - 1 = 1 - 1 = 0

`g(1) = 2|a - 1/2| - 1= 2(1/2) - 1= 1 -1 = 0`

=>f(1) = g(1)

f(2) = (2)^2 - 2 = 4 - 2 = 2

`g(2) = 2|2-1/2| - 1 = 2(3/2)-1 = 3 -1 = 2`

`=> f(2) = g(2)`

:. f(a) = g(a) ∀ a ∈ A

Hence, the functions *f *and *g* are equal.