If f(x) = √x and g(n) = 1 - n², then the domain of fog is -

Steps to Solve

1. Identify the domain of g(n)

   The function ( g(n) = 1 - n^2 ) is a polynomial, which means it is defined for all real numbers. Therefore, the domain of ( g(n) ) is:

   $$ D_g = (-\infty, \infty) $$

2. Evaluate g(n) to find the range for f(x)

   Next, we need to find the range of ( g(n) ). The function ( g(n) ) achieves its maximum value when ( n^2 ) is minimized. The minimum of ( n^2 ) is 0, so:

   $$ g(0) = 1 - 0 = 1 $$

   As ( n ) approaches positive or negative infinity, ( g(n) ) approaches negative infinity. Therefore, the range of ( g(n) ) is:

   $$ R_g = (-\infty, 1] $$

3. Determine the conditions for f(x)

   The function ( f(x) = \sqrt{x} ) is only defined for non-negative values ( x ). Therefore, the input of ( f(x) ), which is ( g(n) ), must satisfy:

   $$ g(n) \geq 0 $$

   This leads us to the inequality:

   $$ 1 - n^2 \geq 0 $$

   Rearranging gives:

   $$ n^2 \leq 1 $$

   Thus, ( n ) must satisfy:

   $$ -1 \leq n \leq 1 $$

4. Combine domains to find the domain of fog

   The domain of the composite function ( (f \circ g)(n) ) is the values of ( n ) for which ( g(n) ) is within the domain of ( f(x) ). Since we found ( -1 \leq n \leq 1 ), we conclude that the domain of ( f \circ g ) is:

   $$ D_{fog} = [-1, 1] $$