range of f(x)=4+(-4x+7)/(x^2+x-2)

Solution

range

f (x) = 4 + \frac{- 4 x + 7}{x ^{2} + x - 2}

f (x) \leq - \frac{2 5}{3} + 2 or \frac{2 5}{3} + 2 \leq f (x) < 4 or f (x) > 4

Interval Notation

(- \infty, - \frac{2 5}{3} + 2] \cup [\frac{2 5}{3} + 2, 4) \cup (4, \infty)

Solution steps

The function range is the combined domain of the inverse functions

Find the inverse function(s) of:

4 + \frac{- 4 x + 7}{x ^{2} + x - 2}

Inverse of

4 + \frac{- 4 x + 7}{x ^{2} + x - 2} : \frac{- x + 9 x ^{2} - 36 x + 16}{2 ( x - 4 )}, \frac{- x - 9 x ^{2} - 36 x + 16}{2 ( x - 4 )}

\frac{- x + 9 x ^{2} - 36 x + 16}{2 ( x - 4 )}, \frac{- x - 9 x ^{2} - 36 x + 16}{2 ( x - 4 )}

Find the domain of each inverse function

Domain of

\frac{- x + 9 x ^{2} - 36 x + 16}{2 ( x - 4 )} : x \leq - \frac{2 5}{3} + 2 or \frac{2 5}{3} + 2 \leq x < 4 or x > 4

Domain of

\frac{- x - 9 x ^{2} - 36 x + 16}{2 ( x - 4 )} : x \leq - \frac{2 5}{3} + 2 or \frac{2 5}{3} + 2 \leq x < 4 or x > 4

Combine the intervals

f (x) \leq - \frac{2 5}{3} + 2 or \frac{2 5}{3} + 2 \leq f (x) < 4 or f (x) > 4

- x^{2} - 6 x + 12

- x^{2} + 2 x + 4

\frac{x ^{3} - x ^{2} - 2 x}{x - 2}

g (x) = x + 8

(\frac{1}{2})^{x - 3}