Let `f:[0, infty) to [0, infty)` and `g: [0, infty) to [0, infty)` be two functions such that f is non-increasing and g is non-decreasing. Let `h(x) = g(f(x)) AA x in [0, infty)`. If `h(0) =0`, then h(2020) is equal to

Options

A. 0
B. 1
C. 1010
D. 2020

Answer

To solve the problem, we need to analyze the functions \( f \) and \( g \) and their properties, as well as the composite function \( h(x) = g(f(x)) \).

### Step-by-Step Solution:

1. **Understanding the Functions:**
- The function \( f: [0, \infty) \to [0, \infty) \) is non-increasing. This means that for any \( x_1 < x_2 \), we have \( f(x_1) \geq f(x_2) \).
- The function \( g: [0, \infty) \to [0, \infty) \) is non-decreasing. This means that for any \( x_1 < x_2 \), we have \( g(x_1) \leq g(x_2) \).

2. **Defining the Composite Function:**
- We define \( h(x) = g(f(x)) \). This means that \( h(x) \) takes the output of \( f(x) \) and applies the function \( g \) to it.

3. **Given Condition:**
- We are given that \( h(0) = 0 \). This implies that \( g(f(0)) = 0 \).

4. **Analyzing \( f(0) \):**
- Since \( f \) is non-increasing and defined on \( [0, \infty) \), the value \( f(0) \) must be some non-negative value (let's denote it as \( a \)). Thus, \( f(0) = a \) where \( a \geq 0 \).

5. **Applying the Condition:**
- From \( h(0) = 0 \), we have \( g(a) = 0 \). Since \( g \) is non-decreasing, if \( g(a) = 0 \), then for any \( x \geq a \), \( g(x) \) must also be greater than or equal to 0. Therefore, \( g(x) \) cannot decrease below 0.

6. **Behavior of \( h(x) \):**
- Since \( f \) is non-increasing, \( f(x) \) will either stay constant or decrease as \( x \) increases. This means that \( f(x) \leq f(0) = a \) for all \( x \geq 0 \).
- Consequently, since \( g \) is non-decreasing and \( g(a) = 0 \), we have \( g(f(x)) \leq g(a) = 0 \) for all \( x \geq 0 \).

7. **Conclusion about \( h(x) \):**
- Since \( h(x) = g(f(x)) \) is always less than or equal to 0 and starts at 0, the only possibility is that \( h(x) \) is constant and equal to 0 for all \( x \geq 0 \).

8. **Finding \( h(2020) \):**
- Therefore, \( h(2020) = 0 \).

### Final Answer:
\[ h(2020) = 0 \]