or \( f(x)=5 x-1 \) and \( g(x)=5 x^{2}-1 \), find the following functions. \( (f \circ g)(x) ; \) b. \( (g \circ f)(x) ; c .(f \circ g)(0) ; \) d. \( (g \circ f)(0) \) a. \( (f \circ g)(x)=\square \) (Simplify your answer.)

To find \( (f \circ g)(x) \), we start with the function \( g(x) \). 1. First, calculate \( g(x) = 5x^2 - 1 \). 2. Next, substitute \( g(x) \) into \( f(x) \): \[ f(g(x)) = f(5x^2 - 1) = 5(5x^2 - 1) - 1 = 25x^2 - 5 - 1 = 25x^2 - 6. \] So, \[ (f \circ g)(x) = 25x^2 - 6. \] Now, let’s work on the other parts! b. For \( (g \circ f)(x) \): 1. Calculate \( f(x) = 5x - 1 \). 2. Substitute \( f(x) \) into \( g(x) \): \[ g(f(x)) = g(5x - 1) = 5(5x - 1)^2 - 1. \] Expanding the square: \[ = 5(25x^2 - 10x + 1) - 1 = 125x^2 - 50x + 5 - 1 = 125x^2 - 50x + 4. \] Thus, \[ (g \circ f)(x) = 125x^2 - 50x + 4. \] c. For \( (f \circ g)(0) \): Calculate \( (f \circ g)(0) = 25(0)^2 - 6 = -6. \) d. For \( (g \circ f)(0) \): Calculate \( (g \circ f)(0) = 125(0)^2 - 50(0) + 4 = 4. \) In summary: a. \( (f \circ g)(x) = 25x^2 - 6 \) b. \( (g \circ f)(x) = 125x^2 - 50x + 4 \) c. \( (f \circ g)(0) = -6 \) d. \( (g \circ f)(0) = 4 \)