How do you simplify and list the restrictions for h (x)= (t^2 - 3t - 4 )/ (t^2 + 9t + 8)?

Sep 17, 2015

$h \left(t\right) = \frac{{t}^{2} - 3 t - 4}{{t}^{2} + 9 t + 8} = 1 - \frac{12}{t + 8}$

with exclusion $t \ne - 1$

Explanation:

$h \left(t\right) = \frac{{t}^{2} - 3 t - 4}{{t}^{2} + 9 t + 8}$

$= \frac{{t}^{2} + 9 y + 8 - 12 t - 12}{{t}^{2} + 9 t + 8}$

$= \frac{{t}^{2} + 9 y + 8}{{t}^{2} + 9 t + 8} - \frac{12 \left(t + 1\right)}{{t}^{2} + 9 t + 8}$

=1 - (12(t+1))/((t+1)(t+8)

$= 1 - \frac{12}{t + 8}$

with exclusion $t \ne - 1$

The value $t = - 1$ is excluded because it results in both the numerator and denominator of $h \left(x\right)$ becoming $0$ and $\frac{0}{0}$ is indeterminate.

Note that $t = - 8$ is not an excluded value from this simplification, since it is equally a singularity of $h \left(x\right)$ and $1 - \frac{12}{t + 8}$.

The domain of both is $\left(- \infty , - 8\right) \cup \left(- 8 , \infty\right)$