Find the Range of the function $f(x) = |x-6|+x^2-1$ 
find the Range of $f(x) = |x-6|+x^2-1$

$$ f(x) = |x-6|+x^2-1 =\left\{ 
\begin{array}{c}
x^2+x-7,& x>0 .....(b) \\ 
5,& x=0 .....(a) \\ 
x^2-x+5,& x<0 ......(c)
\end{array}
\right. 
$$
from eq (b) i got $$f(x)= \left(x+\frac12\right)^2-\frac{29}4 \ge-\frac{29}4$$
and from eq (c) i got $$f(x)= \left(x-\frac12\right)^2+\frac{19}4 \ge\frac{19}4$$
and eq(b) tells me that it also passes through 5 and so generalize all this and found its range is $\left[-\frac{29}4 , \infty\right)$
but the graph says its range is $(5, \infty)$

You should have $x-6<0$, $x-6=0$ and $x-6>0$ respectively. Always look to the entire expression within the absolute value. 
Oh, and another thing: While finding such minimum, you need to check whether it is in the domain. For example, you have $x>0$ (should be $x>6$) for (b), but the minimum is given at $x=-\frac{1}{2}$. 
