Domain of a Composite Function | Step-by-Step Guide with Examples

📚 Learn How to Find the Domain of a Composite Function Step-by-Step! Understanding how to find the domain of a composite function, written as f(g(x)), is essential in algebra, precalculus, and calculus. In this video, we break down:
✔ How to analyze domain restrictions from both the inner and outer functions
✔ Why you must check for additional restrictions
✔ Real examples to help you practice and build confidence

By the end of this lesson, you’ll be able to confidently identify domain restrictions and evaluate composite functions correctly.

🔍 What You’ll Learn in This Video:
✅ What is the Domain of a Composite Function?
✔ f(g(x)) is only defined when g(x) is defined
✔ You must also make sure that f(g(x)) is defined

✅ Steps to Find the Domain of f(g(x)):
✔ Step 1: Find domain restrictions for g(x) (the inner function)
✔ Step 2: Substitute g(x) into f(x) to form the composite
✔ Step 3: Determine any additional restrictions from f(g(x))
✔ Step 4: State the final domain, considering both steps above

✅ Example Problems:
✔ Example 1: f(x) = x/(x + 2), g(x) = 3/x
✔ Example 2: f(x) = x² + 4, g(x) = √x − 2

✅ Common Mistakes to Avoid:
✔ Ignoring domain restrictions of g(x)
✔ Forgetting to evaluate the domain of the composite expression
✔ Assuming the domain of f(g(x)) is the same as g(x)

By the end of this tutorial, you’ll fully understand how to find the domain of a composite function and apply this skill in function analysis and problem solving!

📥 Download Free Guided Notes!
Get guided notes to follow along and practice composite function domains:
👉 https://www.understandthemath.com/alg...

Prerequisite Algebra and Geometry Concepts
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Timestamps
00:00 Introduction
00:22 Steps to Find Domain of a Composite Function
01:54 Example 1
06:32 Example 2

More About Understand The Math
Understand The Math offers clear, accessible, and engaging mathematics instruction for students and lifelong learners. I'm Cheryl Hile, the founder of Understand The Math, with a Ph.D. in Engineering Science and Applied Mathematics and over 25 years of university teaching experience. My goal is to provide accurate, high-quality math instruction through step-by-step explanations and carefully worked-out examples.

Each video includes free guided notes, linked in the description. These notes help students follow along, highlight key concepts, and practice problems, making review easier and more effective.

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