How to find the inverse function for f(x)=the square root of( 4 - 7x / 4 - x).?

Answer 1

So you want to solve for f^(-1)(x), when

f(x) = sqrt [(4 - 7x) / (4 - x)]

Your first step is to make f(x) = y.

y = sqrt [(4 - 7x) / (4 - x)]

Swap the y variables for x and vice versa, then solve for y.

x = sqrt [(4 - 7y) / (4 - y)]

Square both sides,

x^2 = (4 - 7y) / (4 - y)

Now, multiply both sides by (4 - y) to get rid of the fraction.

x^2 (4 - y) = 4 - 7y

Expand the left hand side,

4x^2 - yx^2 = 4 - 7y

Now, move all y terms to the left hand side and everything else to the right hand side.

7y - yx^2 = 4 - 4x^2

Factor out a y, to get

y(7 - x^2) = 4 - 4x^2

And now, isolate the y by dividing appropriately.

y = (4 - 4x^2) / (7 - x^2)

At this point, you make your concluding statement as follows:

f^(-1)(x) = (4 - 4x^2) / (7 - x^2)

However, you're not finished. When the original function has a restriction for x, the inverse will have a restriction for as well.

sqrt [(4 - 7x) / (4 - x)] implies everything inside the square root must be greater than or equal to 0. That is

[(4 - 7x) / (4 - x)] >= 0

Our critical points are the values x that make the left hand side of the inequality equal to 0, or undefined. In this case, (4 - 7x) = 0 implies 4 = 7x, and x = 4/7. x = 4 makes it undefined.

So our critical points are x = {4/7, 4}, and we test a single value for positivity/negativity in these three intervals:

(-infinity, 4/7]
[4/7, 4)
(4, infinity)

For the the interval (-infinity, 4/7], test -1000000 for
(4 - 7x) / (4 - x). We get (positive) divided by (positive), which is positive. We want positive (since the inequality says >= 0). So we keep this interval.

For the interval [4/7, 4), test 1: For (4 - 7x) / (4 - x), we get
(negative) divided by (positive), which is negative. We want positive, so we discard this interval.

For the interval (4,infinity), test 10000000. We'll get negative divided by negative, i.e. positive. Keep this interval.

So our interval where the function is defined is
(-infinity, 4/7] U (4, infinity)

AND our inverse, f^(-1)(x) is equal to

f^(-1)(x) = (4 - 4x^2) / (7 - x^2),
for x in the interval (-infinity, 4/7] U (4, infinity).

************
KEEP IN MIND THE SIGNIFICANCE OF THE DEFINED INTERVAL!!! See the example below as to why.

Ex: Find the inverse of f(x) = sqrt(x).

Proof:

y = sqrt(x)
(swap the variables)
x = sqrt(y)
x^2 = y, or y = x^2

Therefore, f^(-1)(x) = x^2.
THIS IS FALSE. Graphically, inverses are supposed to be symmetric along the line y = x. y = x^2 is a FULL parabola and hence is not the inverse.
But if we look at the original restriction that x is from [0, infinity), this restriction carries over into the inverse.

f^(-1)(x) = x^2, for
x in [0, infinity)

This would graph half of a parabola and would actually be the inverse of y = sqrt(x).

Answer 2

to receive f'(x), basically replace x and y, and are available to a determination for y. f(x) = sqrt (x + 4) x = sqrt (y + 4) x^2 = y + 4 y = x^2 - 4 = (x-2)(x+2) optimal persons have it very very just about actual, yet they have all forgotten one important step: limiting the area. i've got have been given basically been doing statistical math and opportunities presently, so i do no longer undergo in concepts the mindset. i visit regardless of the shown fact that attempt: If I remember properly, you had to receive this sort of f(x) first. as a result, this sort of f(x) = sqrt(x+4) is R = {y | y >= 0}. Then, come across the area of f(x). this is D = {x | x >= -4} Now, we set the area of f '(x) simply by certainty this sort of f(x), and this sort of f(x) simply by certainty the area of f'(x). hence, for f '(x), D = {x | x >= 0}. so which you basically graph the a million/2 of the nice and comfortable parabola (the inverse function) which you have have been given gained. hence, very very final answer: f'(x) = (x-2)(x+2) D: {x | x >= 0}, R: {y | y >= -4} wish it helped.

Answer 3

Set y = f(x) sp that y^2= (4-7x) / (4-x). So that x ( y^2- 7) = 4y^2 - 4, so

x = 4 (y^2 - 1) / ( y^2 - 7).

Answer 4

replace f(x) with y. interchange x and y. solve for y
y= square root of (4-7x / 4-x)
x= square root of (4-7y /4-y)