Consider the polynomial functions p(x)=−2x4+8 and q(x)=5x3−10.
a. (i) Find the domain of each function.
b. (ii) Obtain the range of each function in interval notation.
c. (iii) Solve the equation p(x)=y for a general real number y. Hence, identify all real numbers y for which this equation has no real solution.
d. (iv) Show that the equation q(x)=y has a real solution for every real number y, and explain how the difference in degree parity and leading coefficient sign accounts for the contrasting behaviours observed in parts (iii) and (iv).