SOLUTION: Please help me solve this problem! Let f (x) = 7/5-x A. State the domain B. State the vertical asymptote(s) C. State the horizontal asymptote d. Find the inverse function

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Please help me solve this problem! Let f (x) = 7/5-x A. State the domain B. State the vertical asymptote(s) C. State the horizontal asymptote d. Find the inverse function       Log On


   

Question 150411: Please help me solve this problem!
Let f (x) = 7/5-x
A. State the domain
B. State the vertical asymptote(s)
C. State the horizontal asymptote
d. Find the inverse function of f.
Found 2 solutions by jim_thompson5910, stanbon:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
A)



f%28x%29=%287%29%2F%285-x%29 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=0-5Subtract 5 from both sides


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


x=%28-5%29%2F%28-1%29 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:

which in plain English reads: x is the set of all real numbers except x%3C%3E5

So our domain looks like this in interval notation


note: remember, the parenthesis excludes 5 from the domain

-------------

B)

Vertical Asymptote:
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=0-5Subtract 5 from both sides


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


x=%28-5%29%2F%28-1%29 Divide both sides by -1 to isolate x



x=5 Divide


So the vertical asymptote is x=5

-------------------------------------------------
C)

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.


Horizontal Asymptote:

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=%287%29%2F%285-x%29, we can visually verify our answers:

Graph of y=%287%29%2F%285-x%29%29 with the horizontal asymptote y=0 (blue line) and the vertical asymptote x=5 (green line)

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D)

f%28x%29=7%2F%285-x%29 Start with the given function.


x=7%2F%285-f%28x%29%29 Switch x and f(x)


x%285-f%28x%29%29=7 Multiply both sides by 5-f%28x%29.


5x-x%2Af%28x%29=7 Distribute


-x%2Af%28x%29=7-5x Subtract 5x from both sides.


f%28x%29=%287-5x%29%2F%28-x%29 Divide both sides by -x.


f%28x%29=%28-7%2B5x%29%2F%28x%29 Reduce.


f%28x%29=%285x-7%29%2F%28x%29 Rearrange the terms.


So the inverse function is

Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
Please help me solve this problem!
Let f (x) = 7/(5-x)
---------------
A. State the domain
All Real Numbers except x = 5
-------------------------
B. State the vertical asymptote(s)
x = 5
---------------------------
C. State the horizontal asymptote
y = 0x/-x = 0/-1 = 0
---------------------------
d. Find the inverse function of f.
y = 7/(5-x)
1st: interchange x and y
x = 7/(5-y)
2nd: solve for y
5-y = 7/x
-y = (7/x) -5
y = 5 - (7/x)
y = (5x-7)/x
==============
Cheers,
Stan H.