How to Simplify f(g(x)) Step-by-Step

When we talk about combining functions, we often write it as $f(g(x))$. This is an important idea in algebra and pre-calculus. It lets us mix two functions to make a new one. Here’s a simple way to simplify $f(g(x))$.

Step 1: Identify the Functions

First, we need to know what the functions are. Let’s say we have:

• Function $g(x)$ defined as $g(x) = 2x + 3$.
• Function $f(x)$ defined as $f(x) = x^2 - 5$.

Step 2: Substitute g(x) into f(x)

Next, to find $f(g(x))$, we replace $x$ in $f(x)$ with $g(x)$. This means we will write:

$$f(g(x)) = f(2x + 3).$$

Step 3: Simplify the Expression

Now, we need to put $g(x)$ into $f(x)$. Since $f(x)$ is $f(x) = x^2 - 5$, we will change $x$ to $2x + 3$. This gives us:

$$f(g(x)) = (2x + 3)^2 - 5.$$

Step 4: Expand the Squared Term

Next, we need to work out $(2x + 3)^2$. We use a simple formula for this. The formula says:

$(a + b)^2 = a^2 + 2ab + b^2.$

Here, we let $a = 2x$ and $b = 3$. So:

$$(2x + 3)^2 = (2x)^2 + 2(2x)(3) + 3^2 = 4x^2 + 12x + 9.$$

Step 5: Combine Like Terms

Now, we add this back into our expression for $f(g(x))$:

$$f(g(x)) = 4x^2 + 12x + 9 - 5.$$

If we combine the numbers at the end, we get:

$$f(g(x)) = 4x^2 + 12x + 4.$$

Step 6: Write the Final Expression

So, the final simplified version of $f(g(x))$ is:

$$f(g(x)) = 4x^2 + 12x + 4.$$

Summary

To sum it up, here are the steps to simplify $f(g(x))$:

1. Identify the two functions.
2. Substitute $g(x)$ into $f(x)$.
3. Expand and simplify what you get.
4. Combine any like terms to find your final answer.

This method shows you how to evaluate the combination of functions, which is a key skill in math. You usually practice this by working with different types of functions to see how they behave when mixed together.