1. **State the problem:** Evaluate the composite functions (g \circ f)(-1), (g \circ f)(0), (f \circ g)(-1), and (f \circ g)(4) using the given graphs of y = f(x) and y = g(x). 2. **Recall the definition of composite functions:** - (g \circ f)(x) = g(f(x)) means first find f(x), then plug that result into g. - (f \circ g)(x) = f(g(x)) means first find g(x), then plug that result into f. 3. **Use the given points from the graphs:** - f(-1) = 4, f(0) = 3, f(1) = 2, f(2) = 1, f(3) = 0, f(4) = -1 - g(-1) = 5, g(0) = 7, g(1) = 6, g(2) = 4, g(3) = 3, g(4) = 4 4. **Calculate each part:** (a) (g \circ f)(-1) = g(f(-1)) = g(4) - From f(-1) = 4 - Then g(4) = 4 - So, (g \circ f)(-1) = 4 (b) (g \circ f)(0) = g(f(0)) = g(3) - From f(0) = 3 - Then g(3) = 3 - So, (g \circ f)(0) = 3 (c) (f \circ g)(-1) = f(g(-1)) = f(5) - From g(-1) = 5 - Then f(5) = 0 - So, (f \circ g)(-1) = 0 (d) (f \circ g)(4) = f(g(4)) = f(4) - From g(4) = 4 - Then f(4) = -1 - So, (f \circ g)(4) = -1 **Final answers:** - (g \circ f)(-1) = 4 - (g \circ f)(0) = 3 - (f \circ g)(-1) = 0 - (f \circ g)(4) = -1