2 questions involving Algebra?

Question

I don't understand these...

1) If f(x) = 2x^2 and g(x) = 3x - 5, find:

a) f(g(2))

b) g(f(2))

2) Find the inverse function for each of the following. Be sure to put it in y-form. (Hint: What domain restriction will you need to make on (b) to make it a function?)

a) f(x) = -3x - 2

b) g(x) = 3x^2 - 4

Thanks so much! :)

Answer 1

1) Solve this in steps

g(2) = 3(2) - 5 = 6 - 5 = 1

f(g(2)) = f(1) = 2(1^2) = 2*1 = 2

f(2) = 2(2^2) = 2(4) = 8

g(f(2)) = g(8) = 3(8) - 5 = 24 - 5 = 19

f(g(2)) = 2

g(f(2)) = 19

2)

I solve these by using the form

y = -3x - 2

and solving for x...

x = (y + 2)/3

so the inverse of f(x) is

f-1(y) = (y + 2)/3

------------

g(x) = 3x^2 - 4

y = 3x^2 - 4

x = sqrt( (y+4)/3 )

so...

g-1(y) = sqrt( (y+4)/3 )

Note that in this second function, y must be greater than -4.