Function Problem

Question

Answer 1

y = g(x) = 2x³ - 7

y+7 = 2x³

(y+7)/2 = x³

[(y+7)/2]^1/3 = x

A) x = g^-1(y) = [(y+7)/2]^1/3

B) g^-1(x) = [(x+7)/2]^1/3

Table of values:

x.............g(x)........g^-1(x)

-7............-693.......0

-5 ...........-257.......1

0.............-7...........(7/2)^1/3

1.............-5...........4^1/3

(7/2)^1/3....0 ...........[((7/2)^1/3+7)/2]^1/3

4^{1/3...........}1.............[(4^1/3+7)/2]^1/3

1.62789..1.62790..1.627889

2............9............(9/2)^1/3

9............722........2

Graphs of g(x) and g^-1(x) are reflections of each other along the line y=x.

They intersect near y = x = 1.62789

C) (g^-1 _º g)(x) = [(2x³-7 +7)/2]^1/3 = [(2x³)/2]^1/3 = [x³]^1/3 = x

(g _º g^-1)(x) = 2([(x+7)/2]^1/3)³-7 = 2[(x+7)/2]-7 = (x+7)-7 = x

I notice that it doesn't matter which function you start with.

Since g(x) and g^-1(x) are inverses of each other,

the composition of both functions on x returns the value of x.