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.

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)

1. To find pairs of functions g(x) and h(x) such that f(x) = g(h(x)) for f(x) = (2x + 3)^7, we need to decompose f(x) into a composition of simpler functions.

We can start by expressing f(x) as the composition of two functions, g(x) and h(x). Let's assume that h(x) is the inner function, and its output is used as the input for g(x).

To find g(x) and h(x), we need to reverse engineer the function f(x). Here's how we can do that:

Step 1: Expand the expression (2x + 3)^7 using the binomial theorem. We get:

f(x) = 128x^7 + 672x^6 + 1512x^5 + 1890x^4 + 1512x^3 + 756x^2 + 189x + 27

Step 2: We can see that the constant term 27 is obtained when (2x + 3)^7 = 1. So, we can take g(x) = 2x + 3 and h(x) = 1. Then g(h(x)) = g(1) = 2*1 + 3 = 5.

Step 3: We also observe that the coefficient of x^6 is 672. This can be obtained when (2x + 3)^7 = x^6. So, we can take g(x) = (2x + 3)^7 and h(x) = x^6. Then g(h(x)) = g(x^6) = (2x^6 + 3)^7.

Therefore, we have found two different pairs of functions g(x) and h(x) such that f(x) = g(h(x)):

1. g(x) = 2x + 3, h(x) = 1
2. g(x) = (2x + 3)^7, h(x) = x^6

2. To satisfy the given conditions, let's consider the following functions:

Function 1: f(x) = x^2 + 2x
This function has two zeros at x = 0 and x = -2.

Function 2: g(x) = 1
This function has no zeros since its value is always 1.

Now, let's find the sum of these two functions:

Sum: h(x) = f(x) + g(x) = x^2 + 2x + 1

To determine the number of zeros, we set h(x) to be equal to zero and solve for x:

x^2 + 2x + 1 = 0

By factoring, we get:

(x + 1)(x + 1) = 0

This equation has one zero at x = -1.

Therefore, the functions f(x) = x^2 + 2x and g(x) = 1 satisfy the conditions:

- Function f(x) has two zeros.
- Function g(x) has no zeros.
- The sum of f(x) and g(x) (h(x) = f(x) + g(x)) has one zero.