Question

What is the inverse function of `y=(3x-4)^2`?

Answer 1

`x-9y^2+24y-16=0`

Explanation:

To find the inverse function of a function, you simply switch the `y` with the `x` and the `x` with the `y`. In this case, you could get the inverse function as so:

`x = (3y-4)^2`

You can simplify this expression:

`x=9y^2 - 24y + 16`
`x-9y^2+24y-16=0`

These steps are not necessary, unless explicitly stated.

Answer 2

`" "color(blue)(y=+-((x^(1/2))/3+4/3)`

Explanation:

Given:`" " y=(3x-4)^2`

Square root both sides so that you only have one `x`

`" "y^(1/2)=+-(3x-4)" "`

Note that `y^(1/2)` is another way of writing `sqrt(y)`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(brown)("Consider "y^(1/2)=-(3x-4))`

`" "y^(1/2)=-3x+4`

`" "x=-(y^(1/2))/3+4/3`

Swap the letters round

`" "color(brown)(y=-(x^(1/2))/3 +4/3)`

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(brown)("Consider "y^(1/2)=+(3x-4))`

`" "y^(1/2)+4=3x`

Divide both sides by 3 giving:

`" "(y^(1/2))/3+4/3=x`

Swap the letters round

`" "color(brown)(y=(x^(1/2))/3+4/3)`

;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

`color(blue)("Putting it all together")`

`" "color(blue)(y=+-((x^(1/2))/3+4/3)`

As this is reflected about y=x it means that, for this context, the independent variable is now y and the dependant is x. Thus ,due to the nature of the final condition it is not possible for x to become negative.

Answer 3

This function is not one-one so has no inverse, unless you restrict the domain.

Explanation:

Given `f(x) = y = (3x-4)^2`

Take square roots of both sides to find:

`3x-4 = +-sqrt(y)`

Add `4` to both sides to get:

`3x = 4+-sqrt(y)`

Divide both sides by `3` to get:

`x=(4+-sqrt(y))/3`

This does not define a unique value of `x` for a given `y`, so does not define a function.

If we restrict the domain of the original function to `x in [4/3, oo)` then it does have a well defined inverse:

`f^(-1)(y) = (4+sqrt(y))/3`

with domain `y in [0, oo)`