How to solve fog?

The question "How to solve fog?" is ambiguous. It could refer to either:

1. How to dispel atmospheric fog.
2. How to solve a function composition problem where 'fog' likely represents f(g(x)).

We will address both interpretations.

1. How to Dispel Atmospheric Fog

Dispersing fog in the atmosphere is a challenging problem, often dependent on natural weather processes. However, here are a few approaches:

• Natural Dissipation: Fog naturally dissipates as the temperature rises, or the wind picks up, allowing the moisture to evaporate. This is the most common and environmentally friendly "solution."
• Seeding: In certain conditions, seeding fog with hygroscopic materials (like salt) can encourage water droplets to coalesce and precipitate as drizzle or rain. This is a complex process that is not always effective.
• Heating: In very localized areas, such as airport runways, heating the air can evaporate the fog. This is energy-intensive and not practical on a large scale.
• Ventilation: Mechanical ventilation can be used to displace fog in confined spaces or specific areas, although this method is also limited in scope.

2. How to Solve f(g(x)) (Function Composition)

The abbreviation 'fog' likely refers to the composite function (f o g)(x), which is read as "f of g of x". The reference provided demonstrates how to solve this type of problem.

Understanding Function Composition

Function composition means applying one function to the result of another. In (f o g)(x), you first apply the function g to x, and then apply the function f to the result, g(x).

Example

Let's revisit the example from the provided reference:

• Given: f(x) = 2x and g(x) = x + 1

• Goal: Find (f o g)(x)

Steps:

1. Apply g to x: g(x) = x + 1

2. Apply f to the result of g(x): f(g(x)) = f(x + 1) = 2(x + 1) = 2x + 2

Therefore, (f o g)(x) = 2x + 2. This means for any input x, you first add 1, and then multiply the result by 2.

General Approach

1. Identify the functions: Clearly define f(x) and g(x).

2. Substitute: Replace the 'x' in f(x) with the entire expression for g(x).

3. Simplify: Expand and simplify the resulting expression.

Example 2:

• Let f(x) = x² and g(x) = x - 3

• Then (f o g)(x) = f(g(x)) = f(x - 3) = (x - 3)² = x² - 6x + 9