1. **State the problem:** We need to check the type of the function $$g(x) = 2x^{3} + 3x + 1$$ given the domain $$\{0,1,2,3\}$$ and range $$\{1,6,15,28,35\}$$. 2. **Recall the function type:** The function is a polynomial of degree 3 because the highest power of $$x$$ is 3. 3. **Evaluate the function at each domain value:** - For $$x=0$$: $$g(0) = 2(0)^3 + 3(0) + 1 = 1$$ - For $$x=1$$: $$g(1) = 2(1)^3 + 3(1) + 1 = 2 + 3 + 1 = 6$$ - For $$x=2$$: $$g(2) = 2(2)^3 + 3(2) + 1 = 2(8) + 6 + 1 = 16 + 6 + 1 = 23$$ - For $$x=3$$: $$g(3) = 2(3)^3 + 3(3) + 1 = 2(27) + 9 + 1 = 54 + 9 + 1 = 64$$ 4. **Compare calculated range with given range:** - Calculated range is $$\{1,6,23,64\}$$ - Given range is $$\{1,6,15,28,35\}$$ 5. **Conclusion:** The calculated range does not match the given range. Therefore, either the given range is incorrect or the function does not produce the given range values for the specified domain. 6. **Type of function:** Regardless, $$g(x)$$ is a cubic polynomial function. **Final answer:** The function $$g(x) = 2x^{3} + 3x + 1$$ is a cubic polynomial function. The given range does not correspond to the function values for the given domain.