I need to v'(1), v'(2), and v'(3) when v(x)=f(f(x)).

Question

The picture of the graph given is f(x)=2x from 0<x<2 and f(x)=-2x+8 from 2<x<4.? 

 

 

Answer 1

v(x) = f(f(x))

The function f(x) = 2x when 0 < x < 2

v(x) = f(2x)

= 2(2x)

v(x) = 4x

Differentiating on each side with respect to x.

v'(x) = 4

 

The function f(x) = - 2x + 8 when 2 < x < 4

v(x) = f(f(x))

v(x) = f(- 2x + 8)

= - 2[- 2x + 8] + 8

=  4x - 16 + 8

=  4x - 8

Differentiating on each side with respect to x.

v'(x) =  4

For v'(1), 0 < x < 2

v'(1) = 4

v'(2) does not exist.

For v'(3), 2 < x < 4

v'(3) =  4.