Question

1. The function f(x) = (2x + 3)^7 is the composition of two functions, g(x) and h(x). Find at least two different pairs of functions g(x) and h(x) such that f(x) = g(h(x)).

2. Give an example of two functions that satisfy the following conditions:
- one has 2 zeros
- one has no zeros
- the sum of the functions has 1 zero
Show that your functions satisfy the conditions.

Answer 1

g(x) = x^7
h(x) = 2x+3

g(x) = (2x+1)^7
h(x) = x+1


f(x) = x^2-1
g(x) = x^2+1
(f+g)(x) = 2x^2 (one unique zero, repeated)

f(x) = 3x+x^2
g(x) = 2-x^2
(f+g)(x) = 3x+2 (only a single zero)