How do you find the inverse of #y=x^(2)-6x+4# and is it a function?

Question

How do you find the inverse of  #y=x^(2)-6x+4# and is it a function?

Sophie Adams · Accepted Answer

The inverse fun. of the given fun. does not exist.

Let #y=f(x)=x^2-6x+4=x(x-6)+4.# Hence, #f(0)=4, f(6)=4#, indicating that the fun. #f# is not an injection, i.e., #f# is not #1-1#. The inverse of a given fun. exists #iff# It is bijective (injective = #1-1#, surjective=onto) Therefore, we can not find its inverse function.

Madelyn Anderson · Answer

undefined To find the inverse of $ y = x^2 - 6x + 4 $, follow these steps:

1. Replace $ y $ with $ x $ and $ x $ with $ y $ in the given equation to obtain the inverse function.
2. Solve the resulting equation for $ y $.
3. If the resulting equation is a function, then it is the inverse of the original function.

Starting with the given function $ y = x^2 - 6x + 4 $:

$$ x = y^2 - 6y + 4 $$

Now, solve for $ y $:

$$ x = y^2 - 6y + 4 $$
$$ x - 4 = y^2 - 6y $$
$$ y^2 - 6y = x - 4 $$
$$ y^2 - 6y + 9 = x - 4 + 9 $$
$$ (y - 3)^2 = x + 5 $$
$$ y - 3 = \pm \sqrt{x + 5} $$
$$ y = 3 \pm \sqrt{x + 5} $$

Therefore, the inverse function is $ y = 3 \pm \sqrt{x + 5} $. Since this equation has two possible outputs for each input, it fails the vertical line test, indicating that it is not a function. Therefore, the inverse of $ y = x^2 - 6x + 4 $ is not a function.