Question

1) For the following problem: Let f={(-2,3),(-1,1),(0,0),(1,-1), (2,-3)} and let g-{-3,1),(-1,-2), (0, 2),(2, 2),(3,1)}.Find the following a) f(1) and g-1) b) (gof (1) c) (gofof)(-1) d) (fog)(3)

Answer 1

1. f(1)=-1 and g(-1)=-2.

2. (gof)(1)=-2.

3. (gofof)(-1)=-2

4. (fog)(3)=-1

Explanation:

The given functions are defined as

f={(-2,3),(-1,1),(0,0),(1,-1), (2,-3)}

g={(-3,1),(-1,-2), (0, 2),(2, 2),(3,1)}

1.

The value of function f at x=1 is -1, So, f(1)=-1.

The value of function g at x=-1 is -2, So, g(-1)=-2.

Therefore the value of f(1) is -1 and the value of g(-1) is -2.

2.

`(g\circ f)(1)=g(f(1))`
`[\because (g\circ f)(x)=g(f(x))]`

`(g\circ f)(1)=g(-1)`
`[\because f(1)=-1]`

`(g\circ f)(1)=-2`
`[\because g(-1)=-2]`

Therefore the value of (gof)(1) is -2.

3.

`(g\circ f\circ f)(-1)=(g\circ f)(f(-1))`
`[\because (g\circ f)(x)=g(f(x))]`

`(g\circ f\circ f)(-1)=(g\circ f)(1)`
`[\because f(-1)=1]`

`(g\circ f\circ f)(-1)=-2`
`[\because \text{From part 2}, (g\circ f)(1)=-2]`

Therefore the value of (gofof)(1) is -2.

4.

`(f\circ g)(3)=f(g(3))`
`[\because (f\circ g)(x)=f(g(x))]`

`(f\circ g)(3)=f(1)`
`[\because g(3)=1]`

`(f\circ g)(3)=-1`
`[\because f(1)=-1]`

Therefore the value of (fog)(3) is -1.