Forming the Sum, Difference, Product, and Quotient of Two Functions

Let \(f\) and \(g\) be two functions defined as \[ f( x) =\sqrt{x-1}\qquad\hbox{and}\qquad g( x) =\sqrt{4-x} \] Find the following functions and determine their domain:

1. \(( f+g) ( x)\)
2. \((f-g) ( x)\)
3. \((f\cdot g)( x)\)
4. \(\left( \dfrac{f}{g}\!\right) ( x)\)

Solution The domain of \(f\) is \(\{ x|x\geq 1\} \), and the domain of \(g\) is \(\{ x |x \leq 4\} \).

1. \(( f+g) ( x) =f( x) +g( x) =\sqrt{x-1}+\sqrt{4-x}\). The domain of \(( f+g) ( x) \) is the closed interval \([ 1,4] \).
2. \(( f-g) ( x) =f( x) -g( x) =\sqrt{x-1}-\sqrt{4-x}\). The domain of \(( f-g) ( x) \) is the closed interval \([ 1,4] \).
3. \(( f\cdot g) ( x) =f( x) \cdot g( x)\, {=}\,( \sqrt{x-1}) ( \sqrt{4-x} ) =\sqrt{-x^{2}+5x-4}\). The domain of \(( f\cdot g) ( x) \) is the closed interval \([ 1,4] \).
4. \(\left( \dfrac{f}{g}\right)\ ( x) =\dfrac{ f( x) }{g( x) }=\dfrac{\sqrt{x-1}}{\sqrt{4-x}}=\dfrac{ \sqrt{-x^{2}+5x-4}}{4-x}\). The domain of \(\left( \dfrac{f}{g}\right) ( x) \) is the half-open interval \([1,4)\).