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Manipulating Radical Functions

Manipulating Radical Functions 1.9 - Solution

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Given that f(x)=-5x4f(x)=\text{-}5\sqrt[4]{x} and g(x)=19x4,g(x)=19\sqrt[4]{x}, we will find (f+g)(x)(f+g)(x) and (fg)(x).(f-g)(x). Let's begin with the sum of the functions.

Adding Two Functions

Let's recall the definition we will use. (f+g)(x)=f(x)+g(x)\begin{gathered} (f+g)(x)=f(x)+g(x) \end{gathered} From here, we will substitute the given function rules and simplify.
(f+g)(x)=f(x)+g(x)(f+g)(x)=f(x)+g(x)
(f+g)(x)=-5x4+19x4(f+g)(x)={\color{#0000FF}{\text{-}5\sqrt[4]{x}}}+{\color{#009600}{19\sqrt[4]{x}}}
(f+g)(x)=(-5+19)x4(f+g)(x)=(\text{-}5+19)\sqrt[4]{x}
(f+g)(x)=14x4(f+g)(x)=14\sqrt[4]{x}

Subtracting Two Functions

Let's recall the rule for subtracting functions. (fg)(x)=f(x)g(x)\begin{gathered} (f-g)(x)=f(x)-g(x) \end{gathered} We can proceed in the same way as we did for adding the functions.
(fg)(x)=f(x)g(x)(f-g)(x)=f(x)-g(x)
(fg)(x)=-5x419x4(f-g)(x)={\color{#0000FF}{\text{-}5\sqrt[4]{x}}}-{\color{#009600}{19\sqrt[4]{x}}}
(fg)(x)=(-519)x4(f-g)(x)=(\text{-}5-19)\sqrt[4]{x}
(fg)(x)=-24x4(f-g)(x)=\text{-}24\sqrt[4]{x}