Combining and Composing Functions: A Quick Review

Introduction

In this final review segment we cover the essential operations you can perform on functions: addition, subtraction, multiplication, division, and composition. The goal is to give you a clear, step‑by‑step understanding so you can work with combined functions without needing to revisit the video.

1. Basic Operations on Functions

a) Adding Functions

If (f(x)) and (g(x)) are given, the sum ( (f+g)(x) = f(x)+g(x) ). You simply drop the parentheses and combine like terms.

b) Subtracting Functions

The difference ( (f-g)(x) = f(x)-g(x) ). Remember to distribute the negative sign before simplifying.

c) Multiplying Functions

For the product ( (f\cdot g)(x) = f(x)\times g(x) ) you multiply the expressions and then simplify. No special tricks are required beyond ordinary algebraic expansion.

d) Dividing Functions

The quotient ( \left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} ). Place one function over the other and keep track of the new domain restrictions (e.g., (g(x)\neq 0)).

2. Domain Considerations

When you combine functions, the domain of the resulting function is the intersection of the domains of the original functions. You cannot “fix” a domain problem by combining; you can only make the domain smaller (or keep it the same). For example, if (f(x)=\sqrt{x}) and (g(x)=\frac{1}{x-3}), the combined function (f+g) is defined only for (x\ge 0) and (x\neq 3).

Key points: - Always check each original function for restrictions (square‑roots, even roots, denominators, logarithms, etc.). - After combining, list the common allowed values using the intersection symbol ((\cap)). - Never assume the simplified expression expands the domain; the original restrictions still apply.

3. Function Composition

Composition means inserting one function into another: ( (f\circ g)(x) = f\big(g(x)\big) ).

Steps to Compose

1. Write the outer function (f) and identify its variable (usually (x)).
2. Replace every occurrence of that variable in (f) with the entire expression of the inner function (g(x)).
3. Simplify if possible.

Example

• (f(x)=x^3-4)
• (g(x)=\sqrt{x})

( (f\circ g)(x) = f\big(g(x)\big) = (\sqrt{x})^3-4 = x^{3/2}-4 ).

Multiple Compositions

You can chain more than two functions: ( (f\circ g\circ h)(x) = f\big(g\big(h(x)\big)\big) ).

Example with three functions: - (h(x)=x^3) - (g(x)=\frac{1}{x}) - (f(x)=\sqrt{x})

( (f\circ g\circ h)(x) = f\big(g\big(h(x)\big)\big) = \sqrt{\frac{1}{x^3}} = x^{-3/2} ).

4. Working Backwards

Given a complicated expression, you can often rewrite it as a composition by identifying an “inner” part and an “outer” part. For (u(x)= (x-7)^3), let (g(x)=x-7) and (f(x)=x^3). Then (u = f\circ g).

Conclusion

Combining functions involves straightforward algebraic operations, but the crucial insight is that the domain of the result is always the intersection of the original domains. Composition is simply substitution of one function into another, and the same domain‑intersection rule applies. Mastering these tools lets you manipulate and analyze complex expressions efficiently.

When you add, subtract, multiply, divide, or compose functions, always remember: the new domain is the intersection of the original domains, and composition is just systematic substitution. These principles let you handle any combined‑function problem without extra hassle.