Question 147184
*[Tex \LARGE (f-g)(x)=f(x)-g(x)] Start with the given property



*[Tex \LARGE f(x)-g(x)=\left(2x-5\right)-\left(2-x\right)] Plug in {{{f(x)=2x-5}}} and {{{g(x)=2-x}}}



{{{f(x)-g(x)=2x-5-2+x}}} Distribute.



{{{f(x)-g(x)=3x-7}}} Combine like terms





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*[Tex \LARGE (\frac{f}{g})(x)=\frac{f(x)}{g(x)}] Start with the given property



*[Tex \LARGE \frac{f(x)}{g(x)}=\frac{2x-5}{2-x}] Plug in {{{f(x)=2x-5}}} and {{{g(x)=2-x}}}



{{{2-x=0}}} Set the denominator equal to zero. Remember, dividing by 0 is undefined. So if we find values of x that make the denominator zero, then we must exclude them from the domain.




{{{-x=0-2}}}Subtract 2 from both sides



{{{-x=-2}}} Combine like terms on the right side



{{{x=(-2)/(-1)}}} Divide both sides by -1 to isolate x




{{{x=2}}} Divide






Since {{{x=2}}} makes the denominator equal to zero, this means we must exclude {{{x=2}}} from our domain


So our domain is:  *[Tex \LARGE \textrm{\left{x|x\in\mathbb{R} x\neq2\right}}]


which in plain English reads: x is the set of all real numbers except {{{x<>2}}}


So our domain looks like this in interval notation

*[Tex \Large \left(-\infty, 2\right)\cup\left(2,\infty \right)]


note: remember, the parenthesis <font size=4><b>excludes</b></font> 2 from the domain


If we wanted to graph the domain on a number line, we would get:


{{{drawing(500,50,-10,10,-10,10,
number_line( 500, -8, 12),
blue(arrow(0.2,-7,10,-7)),
blue(arrow(0.2,-6.5,10,-6.5)),
blue(arrow(0.2,-6,10,-6)),
blue(arrow(0.2,-5.5,10,-5.5)),
blue(arrow(0.2,-5,10,-5)),
blue(arrow(-0.2,-7,-10,-7)),
blue(arrow(-0.2,-6.5,-10,-6.5)),
blue(arrow(-0.2,-6,-10,-6)),
blue(arrow(-0.2,-5.5,-10,-5.5)),
blue(arrow(-0.2,-5,-10,-5)),

circle(0,-5.8,0.35),
circle(0,-5.8,0.4),
circle(0,-5.8,0.45),
circle(0,-5.8,0.4),
circle(0,-5.8,0.45)
)}}} Graph of the domain in blue and the excluded value represented by open circle


Notice we have a continuous line until we get to the hole at {{{x=2}}} (which is represented by the open circle).
This graphically represents our domain in which x can be any number except x cannot equal 2