Let h(x) = f(g(x)). Compute h^-1(x)

Question

Need Help.

Let h(x) = f(g(x)). Compute h^-1(x).

These values are given:

f(-2) = -1

f(-1) = 2

f(0) = 1/2

f(1) = 1

f(2) = 3

f(3) = 4

g(-2) = 2

g(-1) = 1

g(0) = 0

g(1) = -1

g(2) = 3

g(3) = -2

f '(-2) = 1

f '(-1) = 4

f ' (0) = 2

f ' (1) = 3

f ' (2) = -1

f ' (3) = -2

g '(-2) = -1

g ' (-1) = 1

g ' (0) = 2

g ' (1) = 3

g ' (2) = 1/2

g ' (3) = 1

Answer 1

h(x) = f( g(x) ) and we are to find the inverse of h(x). So I am going to start a list of the points in h(x). Then at the very last step, we will swap all of the x's and y's to get h^-1(x).

h(x)

Start with f(-2) = -1 f( g(x) ) = -1 so g(x) must = -2, look up which g(x) = -2. And it is when x = 3.

So now we know that f( g(3) ) = -1 This means that our point in h(x) is (3, -1)

(3, -1)

Start with f(-1) = 2 f( g(x) ) = 2 so g(x) = -1 and this happens at g(1) so f( g(1) ) = 2

(1, 2)

Start with f(0) = 1/2 f( g(x) ) = 1/2 so g(x)=0, looking up g(x)=0, find this is at g(0) = 0 f( g(0) ) = 1/2

(0, 1/2)

Starting with f(1) = 1, g(x)=1, find g(-1) = 1 so f( g(-1) ) = 1

(-1, 1)

Starting with f(2) = 3, g(x) = 2, find g(-2) = 2 so f ( g(-2) ) = 3

(-2, 3)

Starting with f(3) = 4 means g(x)=3, find g(2) = 3 so f( g(2) ) = 4

(2, 4)

so h^-1(x) is the points

above with the order changed: h^-1(x) = { (-1, 3), (2, 1), (1/2, 0), (1, -1), (3, -2), (4, 2) }