Find \(f+g\), \(f-g\) and \(f/g\)
Question
Let \(f, g:\mathbb{R}\to\mathbb{R}\) be defined respectively by
\[ f(x)=x+1 \] \[ g(x)=2x-3 \]Find
\(f+g\), \(f-g\) and \(\frac{f}{g}\).
Solution
Given
\[ f(x)=x+1 \]and
\[ g(x)=2x-3 \]Find \(f+g\)
By definition,
\[ (f+g)(x)=f(x)+g(x) \] \[ =(x+1)+(2x-3) \] \[ =3x-2 \]Therefore,
\[ (f+g)(x)=3x-2 \]Find \(f-g\)
By definition,
\[ (f-g)(x)=f(x)-g(x) \] \[ =(x+1)-(2x-3) \] \[ =x+1-2x+3 \] \[ =4-x \]Therefore,
\[ (f-g)(x)=4-x \]Find \(\frac{f}{g}\)
By definition,
\[ \left(\frac{f}{g}\right)(x)=\frac{f(x)}{g(x)} \] \[ =\frac{x+1}{2x-3} \]Since denominator cannot be zero,
\[ 2x-3\ne0 \] \[ x\ne\frac32 \]Therefore,
\[ \left(\frac{f}{g}\right)(x)=\frac{x+1}{2x-3}, \qquad x\ne\frac32 \]