functions math problem?

Question

The function f is one to one. Find its inverse and check your answer. State the domain and the reang if f and f^-1.

f(x) = 4 / x+2

please help me and explain, I have other problems like this :)

thanks!

Answer 1

I always find it helpful to substitute y for f(x).
This makes the equation into:
y = 4/(x+2)

To find the inverse, swap the places of x and y:
x = 4/(y+2)
And then solve for y.
x*(y+2) = 4
y + 2 = 4/x
y = (4/x) - 2

The inverse function is f^-1(x) = (4/x) - 2.

As for domain and range for f(x) = 4/(x+2),
obviously, you can't have a zero denominator, so x is not allowed to be -2. Any other x works, so the domain is all real numbers except x=-2.

As for the range, (x+2) can be positive or negative, and it can be as large as you want, or get arbitrarily close to zero. Accordingly, the fraction 4/(x+2) can be any number except zero.

Hope that helps!

Answer 2

f(x) = 4/(x+2)

To find f inverse, solve for x:

4/f(x) = x + 2

=>

x = 4/f(x) - 2

Switch x and f(x):

f(x) = 4/x - 2

Thus, f inverse = 4/x - 2.

Domain f = {x | x =/ -2} (since you cannot divide by zero)
Domain f inverse = {x | x =/ 0} (same reasoning as above)

Note that:

Domain f = Range f inverse
Domain f inverse = Range f

Answer 3

to find the inverse it is a case of solving for x

ie

x = 4 / (f(x)) -2

thus f(inverse x) = 4/x -2

as for domain x not = 0 range y not = -2

Answer 4

f(x)
y = 4 / x+2
domain : x+2 <> 0 then x <> -2
range : x+2 = 4/y then y<>0

then f^-1
x = 4/y+2
y+2 = 4/x
y = (-2x+4)/x
f^-1(x) = (-2x+4)/x

Domain r-1 = Range r
Range r -1 = Domain r

Df-1 = R - {0}
Rf-1 = R -{2}