Find equation for the inverse of this function: f(x) = (x + 3) / (2x - 4)?

Question

Also the equation for the inverse of:

f(x) = x^2 - 7 is greater than or equal to 0

I don't know if I have the right answer or not. Please show work.

Thank you

Correction: f(x)=x^2-7, x > 0

Answer 1

The inverse is found by switching the x and y and then solving for y. So for f(x) = (x + 3) / (2x - 4) ------ we would have x = (y + 3) / (2y - 4).

Cross multiply to get (2y-4)x = y+3.
Now multiply to get 2xy-4x=y+3.
Rearrange to get 2xy-y=3+4x.
Factor: y(2x-1)=3+4x.
Divide: y = (3+4x)/(2x-1).

The other problem is found the same way: x=y^2-7.
Thus x+7 = y^2
Answer +- Square Root of (x+7).

Answer 2

Equation in first condition

x = f ^-1((x+3)/(2x-4)) , x not equal to 2.

Equation in second condition

x = f ^(x^2-7) , x not equal to square root 7.

Answer 3

For the first function replace f(x) with y.
Then switch x and y. You get x=(y+3)/(2y-4). Cross multiply x(2y-4)=y+3 and then distribute x to get 2xy-4x=y+3.
Subtract y from both sides and add 4x to both sides. 2xy-y=4x+3
Factor out y: y(2x-1)=4x+3
Divide both sides by 2x-1: y=(4x+3)/(2x-1)
Rewrite it: f^-1(x)=(4x+3)/(2x-1)

For the second function start again by replacing
f(x) with y.
Switch x and y: x=y^2-7
Add 7 to both sides: y^2=x+7
Square root both sides: y=+/- sqrt(x+7)
Since you want greater than or equal to zero it will be y=sqrt(x+7).
Rewrite it: f^-1(x)=sqrt(x+7)

Answer 4

f(x) = y => f^-a million(y) = x => 2x³ +5x + 4 = y => x³ + 2.5 x + 2 - y/2 = 0 the discriminant = (2-y/2)² + (4/27) 2.5³ > 0, so certainly we've precisely one genuine answer, so we are able to invert the function for the whole area of genuine numbers for each fee of y, we'd desire to remedy a cubic equation, e.g. y=-3 => x=-a million y=4 => x=0 y=11 => x=a million