Use the functions f and g in C[−1, 1] to find f, g, f, g, and d(f, g) for the inner productf, g = 1f(x)g(x)dx−1.f(x) = 1,    g(x) = 4x2 − 1(a)    f, g (b)    f (c)    g (d)    d(f, g)

The question seems to be asking for the inner product of two functions f and g, the norms of these functions, and the distance between them in the space of continuous functions over the interval [-1, 1] with respect to a specific inner product. The functions are given as f(x) = 1 and g(x) = 4x^2 - 1. The inner product is defined as the integral from -1 to 1 of the product of the two functions.

(a) To find the inner product <f, g>, we compute the integral from -1 to 1 of f(x)g(x) dx. That is, ∫ from -1 to 1 of (1)(4x^2 - 1) dx. Evaluating this integral gives us 0.

(b) To find the norm of f, denoted ||f||, we compute the square root of the inner product <f, f>. That is, sqrt(∫ from -1 to 1 of (1)(1) dx). Evaluating this integral and taking the square root gives us sqrt(2).

(c) To find the norm of g, denoted ||g||, we compute the square root of the inner product <g, g>. That is, sqrt(∫ from -1 to 1 of (4x^2 - 1)(4x^2 - 1) dx). Evaluating this integral and taking the square root gives us sqrt(8/5).

(d) To find the distance between f and g, denoted d(f, g), we compute the norm of the difference between the two functions. That is, ||f - g|| = sqrt(<f - g, f - g>). This simplifies to sqrt(∫ from -1 to 1 of (1 - (4x^2 - 1))^2 dx). Evaluating this integral and taking the square root gives us sqrt(16/15).