Explanation:

1. A linear function has the form $h(x) = mx + c$ , where $m$ is the slope and $c$ is the y-intercept.

2. Calculate the slope ( $m$ ) using the two given points $(x\_1, h(x\_1)) = (4,17)$ and $(x\_2, h(x\_2)) = (6,23)$:

   $m = \dfrac{h(x\_2) - h(x\_1)}{x\_2 - x\_1} = \dfrac{23 - 17}{6 - 4} = \dfrac{6}{2} = 3$ .

   So, the slope is $3$ . The equation is now $h(x) = 3x + c$.

3. Use one of the points to find the y-intercept ( $c$ ):

   Using the point $(4,17)$ : Substitute $x=4$ and $h(x)=17$ into the equation.

   $17 = 3(4) + c$

   $17 = 12 + c$

   $c = 17 - 12$

   $c = 5$ .

4. Write the equation for h(x):

   $h(x) = 3x + 5$ .

5. Verify with the second point $(6,23)$ (optional):

   $h(6) = 3(6) + 5 = 18 + 5 = 23$ . This matches the table.

6. Conclusion:

   The equation that defines h is $h(x)=3x+5$ .