Question

The function f(x) is shown below.

x
f(x)
–6
1
–3
2
2
5
5
3
8
0

If g(x) is the inverse of f(x), what is the value of f(g(2))?
–6
–3
2
5

Answer 1

2

Explanation:

x f(x)

-6 1

-3 2

2 5

5 3

8 0

We will use two point slope form to find function

Formula :
`y-y_1=(y_2-y_1)/(x_2-x_1)(x-x_1)`

`(x_1,y_1)=(-6,1) \\(x_2,y_2)= (-3,2)`

Substitute the values in the formula :

`y-1=(2-1)/(-3+6)(x+6)`

`y-1=(1)/(3)(x+6)`

`3y-3=x+6`

`3y=x+9`

`y=(1)/(3)x+3`

So,
`f(x)=(1)/(3)x+3`

find the inverse of f(x)

`y=(1)/(3)x+3`

Substitute y =x and x = y

`x=(1)/(3)y+3`

`x-3=(1)/(3)y`

`3x-9=y`

So,
`g(x)=3x-9`

Thus the inverse of f(x) =
`g(x)=3x-9`

`f(g(x))=(1)/(3)(3x-9)+3`

Substitute x = 2

`f(g(2))=(1)/(3)(3(2)-9)+3`

`f(g(2))=2`

Hence the value of f(g(2)) is 2

Answer 2

Third option is correct.

Explanation:

The given table is

x f(x)

–6 1

–3 2

2 5

5 3

8 0

If the coordinates of a function f(x) is defined as (x,y), then the coordinates of inverse of f(x) is defined as (y,x).

`f(g(y))=f(x)`
`[\because g(x)=(y,x)]`

`f(g(y))=y`
`[\because f(x)=(x,y)]`

If g(x) is the inverse of f(x), then the value of f(g(y)) is y.

It is given that g(x) is the inverse of f(x), then the value of f(g(2)) is 2.

Therefore third option is correct.