When you want to combine two functions, it’s helpful to follow some easy steps. Combining functions means using one function as the input for another. This is shown as $(f \circ g)(x)$. This notation means you do $g$ first, then $f$ on the result of $g$. Let’s go through the steps together to see how it works:

Step 1: Identify the Functions

First, you need to find the functions you want to combine. For example, let’s use these:

• Function $f(x) = 2x + 3$
• Function $g(x) = x^2$

Step 2: Write the Composition Notation

Next, decide which combination you want to find. You can find either $(f \circ g)(x)$ or $(g \circ f)(x)$. In this example, we’ll find $(f \circ g)(x)$.

Step 3: Substitute the Inner Function

Now, replace $x$ in the function $f(x)$ with the entire function $g(x)$. This means we rewrite $f$ like this:

$$f(g(x)) = f(x^2)$$

Step 4: Evaluate the Outer Function

Next, you will apply the outer function $f$ to what you got from the inner function:

$$f(x^2) = 2(x^2) + 3$$

Step 5: Simplify the Expression

Now, simplify what you have by doing the math:

$$f(x^2) = 2x^2 + 3$$

This result shows the composition $(f \circ g)(x)$.

Step 6: Verify the Composition

It’s always good to check if your answer is right. You can use a number like $x = 2$ to test it.

1. First, calculate $g(2)$: $g(2) = 2^2 = 4$

2. Now find $(f \circ g)(2)$: $f(g(2)) = f(4) = 2(4) + 3 = 8 + 3 = 11$

Step 7: Repeat for the Other Composition (Optional)

If you want, you can do the same steps for $(g \circ f)(x)$:

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

Then, make sure to simplify it as needed.

Conclusion

Combining functions is an important idea in algebra. It helps you understand the relationship between different functions and how they work together. Learning this will be useful in studying more advanced math topics like calculus!