Answer
$\color{blue}{\bf{ 3x^2 - 4x + 3 }; \text{ domain } (-\infty,\infty) }$
$\color{blue}{\bf{ x^2 - 2x - 3 }; \text{ domain } (-\infty,\infty) }$
$\color{blue}{\bf{ 2x^4-5x^3+9x^2-9x }; \text{ domain } (-\infty,\infty) }$
$\color{blue}{\bf{ \dfrac{ 2x^2 - 3x }{ x^2 - x + 3 } }; \text{ domain } (-\infty,\infty) }$
Work Step by Step
We are given the two functions $\bf{f}$ and $\bf{g}$
$\bf{f(x) = 2x^2 - 3x }$ and $\bf{g(x) = x^2 - x + 3 }$
We are asked to find $\bf{( f+g )( x )}$ and its domain:
$( 2x^2 - 3x )+( x^2 - x + 3 ) $
$2x^2+x^2 - 3x - x + 3 $
$\color{blue}{\bf{ 3x^2 - 4x + 3 }}$
$x$ can be any real number so its domain is:
$\color{blue}{\bf{ (-\infty,\infty) }}$
We are asked to find $\bf{( f-g )( x )}$ and its domain:
$( 2x^2 - 3x )-( x^2 - x + 3 )$
$2x^2 - 3x - x^2 + x - 3 $
$2x^2 - x^2 - 3x+ x - 3 $
$\color{blue}{\bf{ x^2 - 2x - 3 }}$
$x$ can be any real number so its domain is:
$\color{blue}{\bf{ (-\infty,\infty) }}$
We are asked to find $\bf{( fg )( x )}$ and its domain:
$( 2x^2 - 3x )( x^2 - x + 3 ) $
$2x^4-2x^3+6x^2-3x^3+3x^2-9x$
$\color{blue}{\bf{ 2x^4-5x^3+9x^2-9x }}$
$x$ can be any real number so its domain is:
$\color{blue}{\bf{ (-\infty,\infty) }}$
We are asked to find $\bf{( \dfrac{f }{ g} )( x)}$ and its domain:
$\color{blue}{\bf{ \dfrac{ 2x^2 - 3x }{ x^2 - x + 3 } }}$
$x$ can be any real number so its domain is:
$\color{blue}{\bf{ (-\infty,\infty) }}$
