SOLUTION: Let f(x) = (2x + 5)/(x - 4). If f^{-1} is the inverse of f, what is f^{-1}(1)?

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Question 1209327: Let
f(x) = (2x + 5)/(x - 4).
If f^{-1} is the inverse of f, what is f^{-1}(1)?
Found 4 solutions by mccravyedwin, ikleyn, greenestamps, math_tutor2020:
Answer by mccravyedwin(409) About Me  (Show Source):
You can put this solution on YOUR website!
This is the graph of f(x)



%22f%28x%29%22%22%22=%22%22%282x+%2B+5%29%2F%28x+-+4%29

Substitute y for f(x)

y%22%22=%22%22%282x+%2B+5%29%2F%28x+-+4%29

Interchange x and y

x%22%22=%22%22%282y+%2B+5%29%2F%28y+-+4%29

Solve for y:

x%28y-4%29%22%22=%22%222y+%2B+5

xy-4x%22%22=%22%222y+%2B+5

xy-2y%22%22=%22%225%2B4x

y%28x-2%29%22%22=%22%225%2B4x

y%22%22=%22%22%285%2B4x%29%2F%28x-2%29

Replace y by f-1(x):

matrix%281%2C2%2Cf%5E%28-1%29%2C%22%28x%29%22%29%22%22=%22%22%285%2B4x%29%2F%28x-2%29

The green graph below is the graph of f-1(x), with the point (1,-9)
circled which shows the answer -9:



matrix%281%2C2%2Cf%5E%28-1%29%2C%22%281%29%22%29%22%22=%22%22%285%2B4%281%29%29%2F%281-2%29=%285%2B4%29%2F%28-1%29=9%2F%28-1%29=-9   <---answer

Edwin

Answer by ikleyn(52879) About Me  (Show Source):
You can put this solution on YOUR website!
.
Let f(x) = (2x + 5)/(x - 4).
If f^{-1} is the inverse of f, what is f^{-1}(1)?
~~~~~~~~~~~~~~~~~~~~~ 

All you need to do to find f^{-1}(1)  is to solve this equation for x

    %282x%2B5%29%2F%28x-4%29 = 1.    <<<---===  the sign is corrected after the notice from @greenestamp.


It can be solved in a few lines

    2x + 5 = x - 4.

    2x - x = -4 - 5

       x   =    -9.     


ANSWEER.  f^{-1}(1) is -9.

Solved.

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

What I am trying to explain is that in this problem you do not need
to restore the inverse function  f^{-1}(x)  explicitly as a function of  x,
as the other tutor does.   It is unnecessary work.

All you need to do to find  f^{-1}(1)  is to solve this equation,  f(x) = 1,  for  x.

In problems of this kind,  it is necessary to restore  f^(-1)(x)  explicitly in two cases:

            (1)   if the problem explicitly asks you about it,

    and/or

            (2)   if the problem asks to calculate  f^(-1)(x)  for several/many values of x.


Then restoring the inverse function is justified from the point of view of effectiveness of your efforts.

Otherwise,  if you are asked to find  f^(-1)(x)  for one single value of  x = c,
it is more effective to solve an equation  f(x) = c  for this single value of  "c".



Answer by greenestamps(13209) About Me  (Show Source):
You can put this solution on YOUR website!


(1) for tutor @ikleyn -- you have a wrong sign in your original equation....

(2) The comments from tutor @ikleyn that there is no need to find the inverse function to answer this question are correct.

(3) In a problem where you are in fact asked to find the inverse of a rational function, it is useful to know that the inverse of

y=%28ax%2Bb%29%2F%28cx%2Bd%29

is

y=%28-dx%2Bb%29%2F%28cx-a%29

(Note the pattern: the "b" and "c" stay where they are; the "a" and "d" switch places and both change sign.)

For this problem, given

y=%282x%2B5%29%2F%28x-4%29

the inverse is

y=%284x%2B5%29%2F%28x-2%29

So

ANSWER: f%5E%28-1%29%281%29=%284%2B5%29%2F%281-2%29=9%2F-1 = -9

Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

Sometimes inverse notation can be a pain to work with. Especially with a keyboard.
I find it's better to introduce another function such as g(x)

Let g(x) be the inverse of f(x)
To find g(x), we first replace f(x) with y.
Then swap x and y and solve for y.

f(x) = (2x + 5)/(x - 4)
y = (2x + 5)/(x - 4)
x = (2y + 5)/(y - 4) ...... x and y swap; from here we isolate y.
x(y-4) = 2y + 5
xy-4x = 2y+5
xy-2y = 5+4x
y(x-2) = 5+4x
y = (5+4x)/(x-2)
g(x) = (5+4x)/(x-2) is the inverse of f(x)

To confirm that f and g are inverses of each other, you should prove that
f( g(x) ) = x and f( g(x) ) = x
are both true equations for all x in the domain.
I'll let the student handle this proof.

Once we figure out the inverse, we can then wrap up the question
g(x) = (5+4x)/(x-2)
g(1) = (5+4*1)/(1-2)
g(1) = (9)/(-1)
g(1) = -9 is the final answer.

This is equivalent to saying f^{-1}(1) = -9 i.e.
But again the -1 exponent notation might be a bit clunky to write out on a keyboard.

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Another approach

The input x maps to the output f(x) when applying the f(x) function.

The inverse goes in reverse of this process.
Computing f^{-1}(1) is the same as asking "what x value gives the output y = f(x) = 1?"

Replace f(x) with 1 and solve for x.
f(x) = (2x + 5)/(x - 4)
1 = (2x + 5)/(x - 4)
x-4 = 2x+5
x-2x = 5+4
-x = 9
x = -9 is the x value input needed to arrive at f(x) = 1

Let's check that claim:
f(x) = (2x + 5)/(x - 4)
f(-9) = (2*(-9) + 5)/(-9 - 4)
f(-9) = (-18 + 5)/(-13)
f(-9) = (-13)/(-13)
f(-9) = 1
This verifies the answer.
You can also use graphing tools like Desmos and GeoGebra to verify.


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Answer: -9