Find \(f+g\), \(f-g\), \(fg\) and \(f/g\)
Question
Let
\[ f:[0,\infty)\to\mathbb{R} \]and
\[ g:\mathbb{R}\to\mathbb{R} \]be defined by
\[ f(x)=\sqrt{x} \] \[ g(x)=x \]Find
\(f+g\), \(f-g\), \(fg\) and \(\frac{f}{g}\).
Solution
Given
\[ f(x)=\sqrt{x} \]and
\[ g(x)=x \]Domains of the Functions
Domain of \(f(x)\):
\[ [0,\infty) \]Domain of \(g(x)\):
\[ \mathbb{R} \]Common domain:
\[ [0,\infty) \]Find \(f+g\)
\[ (f+g)(x)=f(x)+g(x) \] \[ =\sqrt{x}+x \]Domain:
\[ [0,\infty) \]Find \(f-g\)
\[ (f-g)(x)=f(x)-g(x) \] \[ =\sqrt{x}-x \]Domain:
\[ [0,\infty) \]Find \(fg\)
\[ (fg)(x)=f(x)\cdot g(x) \] \[ =\sqrt{x}\cdot x \] \[ =x\sqrt{x} \]Domain:
\[ [0,\infty) \]Find \(\frac{f}{g}\)
\[ \left(\frac{f}{g}\right)(x)=\frac{f(x)}{g(x)} \] \[ =\frac{\sqrt{x}}{x} \]Denominator cannot be zero.
\[ x\ne0 \]Therefore,
\[ \left(\frac{f}{g}\right)(x)=\frac{\sqrt{x}}{x} \]Domain:
\[ (0,\infty) \]