SOLUTION: Let {{{f(x)=7/(5-x)}}} (i) State the domain. (ii) State the vertical asymptote(s). (iii) State the horizontal asymptote. (iv) Find the inverse fu

Question 150025: Let

(i) State the domain.

(ii) State the vertical asymptote(s).

(iii) State the horizontal asymptote.

(iv) Find the inverse function of f. Show work.

Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!

i)

Start with the given function

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.

Subtract 5 from both sides

Combine like terms on the right side

Divide both sides by -1 to isolate x

Divide

Since makes the denominator equal to zero, this means we must exclude from our domain

So our domain is:

which in plain English reads: x is the set of all real numbers except

So our domain looks like this in interval notation

note: remember, the parenthesis excludes 5 from the domain

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ii)
Vertical Asymptote:
To find the vertical asymptote, just set the denominator equal to zero and solve for x

Set the denominator equal to zero

Subtract 5 from both sides

Combine like terms on the right side

Divide both sides by -1 to isolate x

Divide

So the vertical asymptote is

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iii)

Looking at the numerator , we can see that the degree is since the highest exponent of the numerator is . For the denominator , we can see that the degree is since the highest exponent of the denominator is .

Horizontal Asymptote:

Since the degree of the numerator (which is ) is less than the degree of the denominator (which is ), the horizontal asymptote is always

So the horizontal asymptote is

Notice if we graph , we can visually verify our answers:

Graph of with the horizontal asymptote (blue line) and the vertical asymptote (green line)

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iv)

Start with the given function.

Switch x and f(x)

Multiply both sides by .

Distribute

Subtract from both sides.

Divide both sides by .

Reduce.

Rearrange the terms.

So the inverse function is

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