Limit Laws
Operation with Functions
Objectives
- Combine functions using algebraic operations.
- Create a new function by composition of functions.
- Use the limit laws to evaluate the limit of a polynomial or rational function.
- Decompose a composite function into its component’s functions.
Summary
For two functions \(f(x)\) and \(g(x)\) with real number outputs, we define new functions \(f + g, f – g, fg,\) and \(f/g\) by the relation:
- \((f + g) (x) = f(x) + g(x)\)
- \((f – g) (x) = f(x) – g(x)\)
- \((fg)(x) = f(x)g(x)\)
- \((f/g)(x) = f(x) / g(x),\) where \(g(x) ≠ 0\)
When the output of one function is used as the input of another, we call the entire operation a composition of functions. For any input \(x\) and the functions \(f\) and \(g\), this action defines a composite function, which we write as \(f ⸰ g\) such that \((f ⸰ g)(x) = f(g(x))\)
The domain of the composite function \(f ⸰ g\) is all \(x\) such that \(x\) is in the domain of \(g\) and \(g(x)\) is in the domain of \(f\). Remember the product is not the same as the composition of \(f(x)\) and \(g(x)\). \(f(x) g(x) ≠ /f(g(x))\)
The domain of a composite function \(f(g(x))\) is the set of those input \(x\) in the domain of \(g\) for which \(g(x)\) is the domain of \(f\).
If given the function composition \(f(g(x))\), determine its domain by:
- Find the domain of \(g\).
- Find the domain of \(f\)
- Find the inputs of \(x\) in the domain of g which \(g(x)\) is the domain of \(f\). Exclude the inputs \(x\) from the domain of \(g\) for which \(g(x)\) is not in the domain of \(f\). The resulting set is the domain of \(f ⸰ g\).