Calculus 1 Lecture 0.4 | Combining and Composition of Functions

This video, “Calculus 1 Lecture 0.4: Combining and Composition of Functions” by Professor Leonard, is a clear and concise review on how to work with functions by combining and composing them—a crucial skill in calculus.

Key Concepts Explained

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Combining Functions: Add, Subtract, Multiply, Divide ➕

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• When you have two functions $ f(x) $ and $ g(x) $, you can create new functions by:
  • Adding: $ (f + g)(x) = f(x) + g(x) $
  • Subtracting: $ (f - g)(x) = f(x) - g(x) $
  • Multiplying: $ (f \cdot g)(x) = f(x) \times g(x) $
  • Dividing: $ \left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} $, ensuring $ g(x) \neq 0 $
• Example: If $ f(x) = \sqrt{x} $, $ g(x) = x - 3 $,
  • Then $ (f + g)(x) = \sqrt{x} + (x - 3) $.
  • Always be mindful of the domain restrictions arising from the original functions, especially avoiding values where division by zero or square roots of negatives occur.
  • The combined function’s domain is the intersection of the original domains (common inputs valid for both functions).

Domain Considerations and Restrictions ⚠️

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• When combining functions, you cannot improve domains; you can only add new restrictions.
• For example, if $ f(x) $ is undefined for some $ x $, then $ (f + g)(x) $ is also undefined there.
• Always check domain intersections to avoid plugging in invalid inputs.

Function Composition: Substituting One Function Into Another 🔄

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• Composition of functions is written as $ (f \circ g)(x) = f(g(x)) $.
• This means plugging the output of $ g(x) $ directly into $ f $.
• Example:
  • If $ f(x) = x^3 - 4 $ and $ g(x) = \sqrt{x} $,
  • Then $ (f \circ g)(x) = f(g(x)) = f(\sqrt{x}) = (\sqrt{x})^3 - 4 $.
• Note the order matters: $ f \circ g \neq g \circ f $ in general.

Multiple Compositions and Backward Compositions 🔄🔄

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• You can compose more than two functions: e.g., $ f \circ g \circ h $.
• The order is inside-out: calculate $ h(x) $ first, then $ g(h(x)) $, then $ f(g(h(x))) $.
• You can also decompose a function into compositions $ f(g(x)) $, identifying the inner and outer functions.

Professor Leonard’s Tips

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• Use parentheses carefully in compositions to substitute functions correctly.
• Be aware of domains at all steps.
• Practice with both simple combinations and compositions to build intuition.
• Composition is not commutative, so order matters a lot.

This video equips students with the foundational skills necessary to combine and compose functions confidently, preparing them for more advanced calculus concepts where these operations are frequently used. ¹