Question

HELP WITH ALL
Determine the composite functions of the following set of functions

Answer 1

QUESTION 1

i) Given
`f(x)=5x-2` and
`g(x)=x^2`.

`(f\circ g)=f(g(x))`

`(f\circ g)=f(x^2)`

We plug in
`x^2` into
`f(x)=5x-2`

`(f\circ g)=5x^2-2`

ii) Given
`f(x)=5x-2` and
`g(x)=x^2`.

`(g\circ f)=g(f(x))`

`(g\circ f)=g(5x-2)`

We plug in
`5x-2` into
`g(x)=x^2`

`(g\circ f)=(5x-2)^2`

`(g\circ f)=25x^2-10x-10x+4`

`(g\circ f)=25x^2-20x+4`

QUESTION 2

i) Given
`g(x)=x^2-4` and
`h(x)=√(2x+1)`.

`(h\circ g)=h(g(x))`

`(h\circ g)=h(x^2-4)`

We plug in
`x^2-4` into
`h(x)=√(2x+1)`

`(h\circ g)=√(2(x^2-4)+1)`

`(h\circ g)=√(2x^2-8+1)`

`(h\circ g)=√(2x^2-7)`

ii) Given
`g(x)=x^2-4` and
`h(x)=√(2x+1)`.

`(g\circ h)=g(h(x))`

`(g\circ h)=g(√(2x+1))`

We plug in
`\sqrt{2x+1` into
`g(x)=x^2-4`

`(g\circ h)=(√(2x+1))^2-4`

`(g\circ h)=2x+1-4`

`(g\circ h)=2x-3`

QUESTION 3

i) Given
`g(x)=x+5` and
`h(x)=√(x-4)`.

`(g\circ h)=g(h(x))`

`(g\circ h)=g(√(x-4))`

We plug in
`√(x-4)` into
`g(x)=x+5`

`(g\circ h)=√((x-4)+5)`

ii) Given
`g(x)=x+5` and
`h(x)=√(x-4)`.

`(g\circ g)=g(g(x))`

`(g\circ g)=g(x+5)`

We plug in
`x+5` into
`g(x)=x+5`

`(g\circ g)=(x+5)+5}`

`(g\circ g)=x+10}`

QUESTION 4

The given functions are;

`f(x)=-2x+1` and
`g(x)=-x^2`

`f\circ f=f(f(x))`

`f\circ f=f(-2x+1)`

`f\circ f=-2(-2x+1)+1)`

`f\circ f=4x-2+1)`

`f\circ f=4x-1)`

ii)

`g\circ g=g(g(x))`

`g\circ g=g(-x^2)`

`g\circ g=-(x^2)^2`

`g\circ g=-x^4`

QUESTION 5

`f(x)=5x-2;g(x)=x^2;h(x)=-3x`

`h\circ g\circ f=h\circ (g(f(x))`

`h\circ g\circ f=h\circ (g(5x-2))`

`h\circ g\circ f=h\circ ((5x-2)^2)`

`h\circ g\circ f=h\circ (25x^2-20x+4)`

`h\circ g\circ f=h(25x^2-20x+4)`

`h\circ g\circ f=-3(25x^2-20x+4)`

`h\circ g\circ f=-75x^2+60x-12)`

ii.

`f\circ f\circ g=f\circ (f(g(x))`

`f\circ f\circ g=f\circ (f(x^2)`

`f\circ f\circ g=f\circ (5x^2-2)`

`f\circ f\circ g=f(5x^2-2)`

`f\circ f\circ g=5(5x^2-2)-2`

`f\circ f\circ g=25x^2-10-2`

`f\circ f\circ g=25x^2-12`