Understanding Function Composition: Solving for f(g(x))

Today, we're going to learn about function composition in mathematics. Specifically, we will solve the problem of finding f(g(x)) for the given functions fx 2x2 and g(x) 3x1. Function composition is an important concept in algebra and is widely used in various fields, from physics to engineering.

What is Function Composition?

Function composition refers to the process of combining two functions such that the output of one function becomes the input of the other. Mathematically, if we have two functions fx and g(x), the composition f(g(x)) means that we first apply the g(x) function to the input x, and then we apply the fx function to the result of g(x).

Solving for f(g(x))

To solve for f(g(x)) given that fx 2x2 and g(x) 3x1, follow these steps:

Step 1: Substitute g(x) into fx

First, replace every x in fx with g(x).

Given: fx 2x2 and g(x) 3x1

Substitute g(x) into fx to get: 2(3x1)2

Step 2: Simplify the Expression

Now, let's simplify the expression step by step.

1. Compute the inner function first: (3x1)2 9x2

2. Substitute this result back into the expression: 2 * 9x2 18x2

Therefore, f(g(x)) 18x2.

Additional Insight: Using FOIL for Expansion

For a more detailed look into the algebraic operations, let's expand the expression using the FOIL (First, Outer, Inner, Last) method. However, this is not necessary for our current problem.

FOIL stands for a method of multiplying two binomials. If we had a different polynomial, we could use it. For example, if g(x) (3x 1), then applying FOIL to 2(3x 1) * (3x 1) would be:

1. First: (3x) * (3x) 9x2

2. Outer: (3x) * (1) 3x

3. Inner: (1) * (3x) 3x

4. Last: (1) * (1) 1

Adding these together: 9x2 3x 3x 1 9x2 6x 1

Conclusion and Practice

Understanding function composition is crucial for solving more complex mathematical problems. By mastering this concept, you can approach various problems with confidence. If you find this concept confusing, practice with different functions to get better at recognizing patterns and simplifying expressions.

Have a great day, and I hope you've learned something new about function composition!