invertible?

Question

f(x) = X ^2 -3

Is it invertible and if so what is its inverse?
I say Not invertible but I want to double check bc I am not good at these.

Along with this one

F (x) = 2x / x - 7

I didn't think that was invertible either....

Answer 1

f(x) = x^2 - 3

let y = x^2 - 3

so y + 3 = x^2

so +/- sqrt (y + 3) = x

I'll let someone else do the other one.

Answer 2

It depends. f:R->R is not invertible. For example, f(-2) = f(2), and there are no values of x so that f(x) = -5. But f:R>=0:->R>=-3 is invertible (it means: the domain set are real numbers greater of equal to zero, the codomain is the set of real numbers greater or equal to -3).

y= f(x) = x^2-3, then x = sqrt(y+3)

f^(-1)(y) = sqrt(y+3), can be calculated only if y>=-3, and only gives positive values.

F(x) = 2x/(x-7) is not invertible as a function of R in R, but it is invertible as a function of R-{7} in R-{2}

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

Answer 3

We need to know the domain for the function before we can decide whether the function is invertible ON that domain .Perhaps there was more to these questions?

For example, the first function is not invertible as a function defined on the reals, but it is invertible as a function defined on the positive reals.

Answer 4

A function is invertible if it is one-one and onto.
Given function is
f(x) = X ^2 -3
for one-one means f(a)=f(b) implies a=b
Let f(a)=f(b)
implies a ^2 -3=b ^2 -3
implies a ^2=b ^2
implies a=plus minus b
implies function is not one-one
hence not invertible.

Answer for your second Question:
Given function is F (x) = 2x / x - 7
Step.1.To check for one-one means f(a)=f(b) implies a=b
Let f(a)=f(b)
implies 2a / a - 7=2b / b - 7
implies 2a(b-7) = 2b(a-7)
implies 2ab-14a = 2ab - 14b
implies a=b
implies function is one-one
Step.2. Onto
given function is not onto

as if we write given function in terms of x
i.e. x= 7y/ (y-2)
here we see corresponding to y=2 there does'nt exist real value of x
hence it is not invertible.

Answer 5

the first is invertible but the inverse is only a function if the domain is restricted.
x = y^2 - 3
x+3 = y^2
sqrt(x+3) = y
f^-1(x) = sqrt(x+3)

The second is harder but needs no restriction

x = 2y/(y-7) [multiply by (y-7)]
x(y-7) = 2y [distribute]
xy - 7x = 2y [add 7x and subtract 2y]
xy - 2y = 7x [factor]
y(x-2) = 7x [divide by (x-2)]
y = 7x/(x-2)

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

Answer 6

let g(y) = SQRT(y+3)

Then (g o f)(x) = SQRT( X^2 -3 + 3) = SQRT(X^2) = X

Here, (g o f)(x) = x for any positive value of x but not for negative values (or vice versa, depending on how you define SQRT). (g o f ) is NOT equivalent to the identity function on the ENTIRE domain.