Question

discuss the continuity of the composite function f o g. f(t)=t^2 g(x,y)=6x - 4y o continuous except at (0,0) o continuous everywhere o continuous everywhere y < 3/2x o continuous everywhere y ≠ 3/2x o continuous everywhere y > 3/2x

Explanation:

Step1: Recall continuity of composite functions

For a composite function \(f\circ g\), if \(g(x,y)\) is continuous at a point \((a,b)\) and \(f(t)\) is continuous at \(g(a,b)\), then \(f\circ g\) is continuous at \((a,b)\). The function \(f(t)=t^{2}\) is a polynomial function and is continuous for all real - valued \(t\). The function \(g(x,y)=6x - 4y\) is a linear function and is continuous for all \((x,y)\in\mathbb{R}^{2}\).

Step2: Determine continuity of \(f\circ g\)

Since \(f(t)\) is continuous for all \(t\in\mathbb{R}\) and \(g(x,y)\) is continuous for all \((x,y)\in\mathbb{R}^{2}\), the composite function \(f\circ g(x,y)=f(g(x,y))=(6x - 4y)^{2}\) is continuous everywhere.

Answer:

continuous everywhere