Find the Range of this function

Question

I also was having trouble finding the range of this equation too.

Find the range

Problem: x²- 4 / x²+ 5x + 6

I did the inverse switching the x and y and got R: (-inf, 1)U(1, inf)

My work: x = y²- 4 / y²+ 5y + 6

x = (y+2)(y-2)/(y+3)(y+2) *Y+2 cancel out

x = (y-2)/(y+3)

(y+3)x = y-2

xy+3x = y-2

xy - y = -2-3x

y(x-1) = -2-3x

y= -2-3x/ x-1

And I took the bottom number as that it could not equal positive 1

Please let me know if I did something wrong.

Answer 1

try another way (instead of inverse domain is original function range)

other way is to graph original and write down range.

function has point disconnect at x=-2 (domain) y=-4 (range)

otherwise f(x)=1+(-5/(x+3) which is an inverted hyperbola having asymptote of y=+1 (range)

range is (-infinity, point discontinuity at y=-4), (-4, asymptote at +1)), (+1,+infiinity)

If x=-2 y=-4 point defined for original function then range is (-infinity,+1), (+1,+infinity)

Answer 2