find the functions.

Question

1) If f(x)=x^2 and g(x)=x+2, find f[g(x)]
2) If f(x)=x^3+4 and g(x)=x+3, find [g•f](2)
3) Find the inverse of the function f(x)=4x+1.

Answer 1

Given functions are f (x ) = x ^2  and g (x ) = x +2

1) f  [ g (x )]

Substitute the expression for functioning g  (in this case x +2 ) for g (x ) in the composition.

= f  (x + 2)

Now substitute this expression (x + 2) in to function f  in place of the x  value.

= (x + 2)^2

= x ^2 + 4x + 4

f  [ g (x )] = x ^2 + 4x + 4.

 

2) Given functions are f (x ) = x ^3+4  and g (x ) = x + 3

To find [ g  . f  ](2) 

g . f  = g (x ) . f (x )

= (x +3)(x ^3+4)

= x ^4 + 4x + 3x ^3 + 12

[g  . f  ](2) = (2)^4 + 4(2)+ 3(2) ^3 + 12

= 16 + 8 + 3(8) +12

= 16 + 8 + 24 + 12

[ g . f  ](2) = 60.

 

3) Given function f (x ) = 4x + 1.

Replace f(x) by y.

y = 4x +1

Interchange the roles of x and y, and solve for y.

x = 4y +1

Subtract 1 from each side.

x -1 = 4y

Divide each side by 4.

(x - 1)/4 = y.

Replace y by f ^-1(x).

Inverse of given function is f ^-1(x)  = (x  - 1)/4.