# Find f(g(x)) and g(f(x)) and determine whether each pair of functions f and g are inverses of each other.1) f(x) = 3x+8 and g(x) = x-8/3   2) f(x) = 3/x-4 and g(x) = 3 divided x.... +4 Please...

Find f(g(x)) and g(f(x)) and determine whether each pair of functions f and g are inverses of each other.

1) f(x) = 3x+8 and g(x) = x-8/3

2) f(x) = 3/x-4 and g(x) = 3 divided x.... +4

Please explain these. My teacher is terrible and didn't explain how to do these correctly.

sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted on

You need to compose the functions f(x) and g(x) such that:

`(fog)(x) = f(g(x))`

You should substitute g(x) for x in equation of f(x):

`f(g(x)) = 3*g(x) + 8 => f(g(x)) = 3*(x-8)/3 + 8 = x - 8 + 8 = x`

You need to compose the functions f(x) and g(x) such that:

`(gof)(x) = g(f(x)) `

You should substitute f(x) for x in equation of g(x):

`g(f(x)) = (f(x) - 8)/3 => g(f(x)) = (3x + 8 - 8)/3 => g(f(x)) = 3x/3`

`g(f(x)) = x`

Since the functions f(x) and g(x) have the property f(g(x)) = g(f(x)) = x, then f(x) and g(x) are inverses to each other.

2) You need to compose the functions f(x) and g(x) such that:

`(fog)(x) = f(g(x))`

You should substitute g(x) for x in equation of f(x):

`f(g(x)) = 3*g(x)-4 => f(g(x)) = 3(3/(x+4))- 4= (9 - 4x - 16)/(x+4)`

`f(g(x)) = (-7-4x)/(x+4)`

Notice that composing f(x) and g(x) does not yield x, hence, g(x) is not the inverse of g(x).