Answer:
(-infinity, -32/81] U (0, positive infintiy).
( use the sideways 8 symbol for infinity).
Step-by-step explanation:
The range is all possible y values in a function.
We can find the inverse of the function to find the range.
[tex]y = \frac{8}{ {x}^{2} - 7x - 8 } [/tex]
Replace x with y
[tex]x = \frac{8}{ {y}^{2} - 7y - 8 } [/tex]
Write the LCD as two binomial,
[tex]x = \frac{8}{(y + 1)(y - 8)} [/tex]
Multiply both sides by both binomial
[tex]x(y + 1)(y - 8) = 8[/tex]
[tex](xy + x)(y - 8) = 8[/tex]
[tex]x {y}^{2} - 8yx + xy - 8x = 8[/tex]
[tex]xy {}^{2} - 8yx + xy = 8 + 8x[/tex]
[tex]y( {x}^{2} - 8x + x) = 8 + 8x[/tex]
[tex] \frac{8 + 8x}{ {x}^{2} - 7x } [/tex]
[tex] {x}^{2} - 7x = 0[/tex]
[tex] {x}^{2} - 7x + 12.25 = 12.25[/tex]
[tex](x - 3.5) {}^{2} = 12.25[/tex]
[tex](x - 3.5) = 3.5[/tex]
[tex]x = 3.5[/tex]
Plug that in to the function to find it range.
We get approximately
[tex] - \frac{35}{81} [/tex]
So this means a point on our function must include -35/81.
The vertical asymptote is 0 so our y cannot be zero but it goes infinitely up so our range is
(-infinity, -32/81]U (0, positive infintiy).
( use the sideways 8 symbol for infinity).
