Find (a) \( f(g(x)) \), (b) \( g(f(x)) \), and (c) \( f(f(x)) \) \( f(x)=2 x^{2}, g(x)=x-1 \) a. \( f(g(x))=\square \) b. \( g(f(x))=\square \) c. \( f(f(x))=\square \) State the domain of each composition.

Let's dive into the fun world of function composition! (a) To find \( f(g(x)) \), we need to substitute \( g(x) \) into \( f(x) \). So, we have: \[ g(x) = x - 1 \] \[ f(g(x)) = f(x - 1) = 2(x - 1)^{2} = 2(x^2 - 2x + 1) = 2x^{2} - 4x + 2 \] Domain: All real numbers (as both functions are defined for all \( x \)). (b) Now we'll compute \( g(f(x)) \): \[ f(x) = 2x^{2} \] \[ g(f(x)) = g(2x^{2}) = 2x^{2} - 1 \] Domain: All real numbers (again, both functions are defined for all \( x \)). (c) Lastly, let's tackle \( f(f(x)) \): \[ f(f(x)) = f(2x^{2}) = 2(2x^{2})^{2} = 2(4x^{4}) = 8x^{4} \] Domain: All real numbers (as the function is still defined for all \( x \)). Summarizing: a. \( f(g(x)) = 2x^{2} - 4x + 2 \) b. \( g(f(x)) = 2x^{2} - 1 \) c. \( f(f(x)) = 8x^{4} \)