Question

Given that f(x) = 6x - 5 g(x) = 3x + 4 and h(x) = 4x – 6

2

Find:-

i) g(-2) = ii) g[h(x)] = iii) f[g(2)] =




iv) gᴏh(2) vi) h-1(11

Answer 1

(i)
`g(-2)=-2`

(ii)
`g[h(x)]=12x-14`

(iii)
`f[g(2)]=55`

(iv)
`(g\circ h)(2)=10`

(v)
`h^(-1)(11)=(17)/(4)`

Explanation:

The given functions are

`f(x)=6x-5`

`g(x)=3x+4`

`h(x)=4x-6`

(i) Find g(-2).

Substitute x=-2 in g(x).

`g(-2)=3(-2)+4\Rightarrow -6+4=-2`

(ii) Find g[h(x)]

`g[h(x)]=g(4x-6)`
`(h(x)=4x-6)`

`g[h(x)]=3(4x-6)+4`
`(g(x)=3x+4)`

`g[h(x)]=12x-18+4`

`g[h(x)]=12x-14`

(iii) Find f[g(2)].

`f[g(2)]=f[3(2)+4]`
`(g(x)=3x+4)`

`f[g(2)]=f(6+4)`

`f[g(2)]=f(10)`

`f[g(2)]=6(10)-5`
`f(x)=6x-5`

`f[g(2)]=55`

iv) Find gᴏh(2).

`(g\circ h)(2)=g[h(2)]`

`(g\circ h)(2)=12(2)-14` (From part (ii) we get
`g[h(x)]=12x-14)`

`(g\circ h)(2)=10`

(v) Find
`h^(-1)(11)`

First find
`h^(-1)(x)`.

`h(x)=4x-6`

`y=4x-6`

`x=4y-6`

`x+6=4y`

Divide both sides by 4.

`(x+6)/(4)=y`

`h^(-1)(x)=(x+6)/(4)`

Substitute x=11 in the inverse function.

`h^(-1)(11)=(11+6)/(4)`

`h^(-1)(11)=(17)/(4)`