Let the function $f : [ 1 , \infty ) \rightarrow \mathbb { R }$ be defined by $$f ( t ) = \left\{ \begin{array} { c c } ( - 1 ) ^ { n + 1 } 2 , & \text { if } t = 2 n - 1 , n \in \mathbb { N } , \\ \frac { ( 2 n + 1 - t ) } { 2 } f ( 2 n - 1 ) + \frac { ( t - ( 2 n - 1 ) ) } { 2 } f ( 2 n + 1 ) , & \text { if } 2 n - 1 < t < 2 n + 1 , n \in \mathbb { N } . \end{array} \right.$$ Define $g ( x ) = \int _ { 1 } ^ { x } f ( t ) d t , x \in ( 1 , \infty )$. Let $\alpha$ denote the number of solutions of the equation $g ( x ) = 0$ in the interval $( 1,8 ]$ and $\beta = \lim _ { x \rightarrow 1 + } \frac { g ( x ) } { x - 1 }$. Then the value of $\alpha + \beta$ is equal to $\_\_\_\_$ .
Let the function $f : [ 1 , \infty ) \rightarrow \mathbb { R }$ be defined by
$$f ( t ) = \left\{ \begin{array} { c c } ( - 1 ) ^ { n + 1 } 2 , & \text { if } t = 2 n - 1 , n \in \mathbb { N } , \\ \frac { ( 2 n + 1 - t ) } { 2 } f ( 2 n - 1 ) + \frac { ( t - ( 2 n - 1 ) ) } { 2 } f ( 2 n + 1 ) , & \text { if } 2 n - 1 < t < 2 n + 1 , n \in \mathbb { N } . \end{array} \right.$$
Define $g ( x ) = \int _ { 1 } ^ { x } f ( t ) d t , x \in ( 1 , \infty )$. Let $\alpha$ denote the number of solutions of the equation $g ( x ) = 0$ in the interval $( 1,8 ]$ and $\beta = \lim _ { x \rightarrow 1 + } \frac { g ( x ) } { x - 1 }$. Then the value of $\alpha + \beta$ is equal to $\_\_\_\_$ .