Suppose that the functions \( f \) and \( g \) are defined as follows. \[ f(x)=\frac{7}{x}, x \neq 0 \] \[ g(x)=x^{2}-1 \] Find the compositions \( f \circ f \) and \( g \circ g \). Simplify. your answers as much as possible. (Assume that your expressions are defined for all \( x \) in the domain of the composition. You do not have to indicate the domain.) \( (f \circ f)(x)=\square \) \( (g \circ g)(x)=\square \)

To find the compositions \( f \circ f \) and \( g \circ g \), we will calculate each composition step by step. First, we compute \( (f \circ f)(x) \): \[ f(f(x)) = f\left(\frac{7}{x}\right) = \frac{7}{\frac{7}{x}} = \frac{7x}{7} = x. \] Thus, \[ (f \circ f)(x) = x. \] Next, we calculate \( (g \circ g)(x) \): \[ g(g(x)) = g(x^2 - 1) = (x^2 - 1)^2 - 1. \] Now, expanding this gives: \[ (x^2 - 1)^2 = x^4 - 2x^2 + 1. \] So, \[ g(g(x)) = x^4 - 2x^2 + 1 - 1 = x^4 - 2x^2. \] Thus, \[ (g \circ g)(x) = x^4 - 2x^2. \] Finally, we can summarize the results: \[ (f \circ f)(x) = x, \] \[ (g \circ g)(x) = x^4 - 2x^2. \]