How do you find the inverse of y=((x^2)-4)/xy=(x2)−4x and is it a function?
1 Answer
Explanation:
Graph the original function:
graph{(x^2-4)/x [-41.1, 41.1, -20.54, 20.55]}
Since it does not pass the horizontal line test, its inverse will not be a function.
To find its inverse still, flip the
y=(x^2-4)/x" "=>" "x=(y^2-4)/yy=x2−4x ⇒ x=y2−4y
Multiply both sides by
xy=y^2-4xy=y2−4
Rearrange to get both terms with
y^2-xy=4y2−xy=4
Now, add
y^2-xy+x^2/4=x^2/4+4y2−xy+x24=x24+4
Note that
(y-x/2)^2=(x^2+16)/4(y−x2)2=x2+164
Take the square root of both sides. Note that we will take the positive and negative versions of this -- this will actually create two separate functions that cannot act as a function on their own since together they break the vertical line test.
y-x/2=(+-sqrt(x^2+16))/2y−x2=±√x2+162
y=(x+-sqrt(x^2+16))/2y=x±√x2+162
Graphed, this should be a reflection of the graph of
graph{(y-(x+sqrt(x^2+16))/2)(y-(x-sqrt(x^2+16))/2)=0 [-52.02, 52.02, -26, 26.02]}
Note that the top line is the graph of
