Question

Consider the following piecewise defined functions:

{ x+1 x < 0
f(x) = { -2 x < 3 g(x) = { x² 0 ≤ x < 4
{ x x ≥ 3 { 3 x ≥ 4

h(x) = { 0 x < 5 k(x) = { 0 x < 2
{ 1 x ≥ 5 { 1 x ≥ 2
Determine the following (your answer should be another piecewise defined function:
(a) f(x)+g(x)
(b) g(x)⋅h(x)
(c) g(x)⋅(1−k(x))
(d) (x²+eˣ)⋅k(x)−(x²+eˣ)⋅h(x)

Answer 1

The student's question requires finding new piecewise functions from provided operations on given functions. Without the complete functions, the calculation cannot be done.

Step-by-step explanation:

The student has asked to consider four piecewise defined functions f(x), g(x), h(x), and k(x), and to determine new piecewise functions based on the operations provided. The operations include addition, multiplication, and the combination of function values with constants. Specifically, the student is supposed to find f(x)+g(x), g(x)⋅h(x), g(x)⋅(1−k(x)), and (x²+eˣ)⋅k(x)−(x²+eˣ)⋅h(x). Due to the nature of the question and its reliance on specific functions which are not fully given, a full answer to the student's query cannot be completed. However, when dealing with piecewise functions, always remember to respect the domain restrictions when performing operations.