Consider the following three functions: g(x, y)= 4x - 3y + 21 H(x, y, z)= 13x² + y + 3z I(z)= z Which of the following is true regarding the linearity of the functions? A)none of the functions is linear B)all three functions are linear C)h is linear, but g and i are not linear D)i is linear, but g and h are not linear E)g and i are linear, but h is not linear

Question

Quizplus · Accepted Answer

A function is linear if it satisfies two conditions: 1) Homogeneity: f(ax, ay) = a f(x,y) 2) Additivity: f(x1 + x2, y1 + y2) = f(x1,y1) + f(x2,y2) g(x,y) = 4x - 3y + 21 is not linear because it doesn't satisfy the first condition of homogeneity. H(x,y,z) = 13x² + y + 3z is not linear because it doesn't satisfy the second condition of additivity. I(z) = z is linear because it satisfies both homogeneity and additivity, i.e., I(ax) = aI(x) and I(x1 + x2) = I(x1) + I(x2). Therefore, option E is the correct answer.

Consider the Following Three Functions