Question 150025

i)




{{{f(x)=(7)/(5-x)}}} Start with the given function



{{{5-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=0-5}}}Subtract 5 from both sides



{{{-x=-5}}} Combine like terms on the right side



{{{x=(-5)/(-1)}}} Divide both sides by -1 to isolate x




{{{x=5}}} Divide






Since {{{x=5}}} makes the denominator equal to zero, this means we must exclude {{{x=5}}} from our domain


So our domain is:  *[Tex \LARGE \textrm{\left{x|x\in\mathbb{R} x\neq5\right}}]


which in plain English reads: x is the set of all real numbers except {{{x<>5}}}


So our domain looks like this in interval notation

*[Tex \Large \left(-\infty, 5\right)\cup\left(5,\infty \right)]


note: remember, the parenthesis <font size=4><b>excludes</b></font> 5 from the domain


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ii)

<b> Vertical Asymptote: </b>

To find the vertical asymptote, just set the denominator equal to zero and solve for x


{{{5-x=0}}} Set the denominator equal to zero



{{{-x=0-5}}}Subtract 5 from both sides



{{{-x=-5}}} Combine like terms on the right side



{{{x=(-5)/(-1)}}} Divide both sides by -1 to isolate x




{{{x=5}}} Divide



So the vertical asymptote is {{{x=5}}}


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iii)


Looking at the numerator {{{7}}}, we can see that the degree is {{{0}}} since the highest exponent of the numerator is {{{0}}}. For the denominator {{{5-x}}}, we can see that the degree is {{{1}}} since the highest exponent of the denominator is {{{1}}}.



<b> Horizontal Asymptote: </b>


Since the degree of the numerator (which is {{{0}}}) is less than the degree of the denominator (which is {{{1}}}), the horizontal asymptote is always {{{y=0}}}


So the horizontal asymptote is {{{y=0}}}




Notice if we graph {{{y=(7)/(5-x)}}}, we can visually verify our answers:


{{{drawing(500,500,-10,10,-10,10,
graph(500,500,-10,10,-10,10,(7)/(5-x)),
blue(line(-20,0,20,0)),
green(line(5,-20,5,20))
)}}} Graph of {{{y=(7)/(5-x))}}}  with the horizontal asymptote {{{y=0}}} (blue line)  and the vertical asymptote {{{x=5}}}  (green line)


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iv)


{{{f(x)=7/(5-x)}}} Start with the given function.



{{{x=7/(5-f(x))}}} Switch x and f(x)



{{{x(5-f(x))=7}}} Multiply both sides by {{{5-f(x)}}}.



{{{5x-x*f(x)=7}}} Distribute



{{{-x*f(x)=7-5x}}} Subtract {{{5x}}} from both sides.



{{{f(x)=(7-5x)/(-x)}}} Divide both sides by {{{-x}}}.



{{{f(x)=(-7+5x)/(x)}}} Reduce.



{{{f(x)=(5x-7)/(x)}}} Rearrange the terms.



So the inverse function is *[Tex \LARGE f^{-1}\left(x\right)=\frac{5x-7}{x}]