Problem 3 - College Algebra and Calculus: An Applied Approach

Question

Find the function values. \(f(x, y)=4-x^{2}-4 y^{2}\) (a) \(f(0,0)\) (b) \(f(0,1)\) (c) \(f(2,3)\) (d) \(f(1, y)\) (e) \(f(x, 0)\) (f) \(f(t, 1)\)

Step-by-Step Solution

Answer

(a) The value of the function at (0,0) is 4. (b) The value of the function at (0,1) is 0. (c) The value of the function at (2,3) is -25. (d) For (1, y), the function can be written as 3-4y^{2}. (e) For (x, 0), the function can be written as 4-x^{2}. (f) For (t, 1), the function can be written as -t^{2}.

Step 1: Find the function value for (0,0)

To find the function value at point (0,0), plug x=0 and y=0 into the function: \(f(0,0)=4-(0)^{2}-4*(0)^{2}\) which simplifies to \(f(0,0)=4\)

Step 2: Find the function value for (0,1)

Plug x=0 and y=1 into the function: \(f(0,1)=4-(0)^{2}-4*(1)^{2}\), which simplifies to \(f(0,1)=0\)

Step 3: Find the function value for (2,3)

Plug x=2 and y=3 into the function: \(f(2,3)=4-(2)^{2}-4*(3)^{2}\), which simplifies to \(f(2,3)=-25\)

Step 4: Find the function value for (1, y)

Plug x=1 into the function, keeping y as a variable: \(f(1,y)=4-(1)^{2}-4*y^{2}\), which can be written as \(f(1,y)=4-1-4y^{2}=3-4y^{2}\)

Step 5: Find the function value for (x, 0)

Plug y=0 into the function, keeping x as a variable: \(f(x,0)=4-x^{2}-4*(0)^{2}=4-x^{2}\)

Step 6: Find the function value for (t, 1)

Plug x=t and y=1 into the function: \(f(t,1)=4-(t)^{2}-4*(1)^{2}\), which simplifies to \(f(t,1)=4-t^{2}-4=0-t^{2}\)