Begin with the innermost function. You are given the value of g(x): g(x) = 2x. Substitute this expression for x in the equation f(x):

f(x) = 4 – 2x²

f(2√x) = 4 – 2(2√x)²

f(2√x) = 4 – 2(4x)

f(2√x) = 4 – 8x

Therefore, f(g(x)) = 4 – 8x

Begin with the innermost function. You are given the value of f(x): f(x) = 6x + 4. Substitute this expression for x in the equation g(x):

g(x) = x² – 1

g(6x + 4) = (6x + 4)² – 1

g(6x + 4) = 36x² + 24x + 24x + 16 – 1

g(6x + 4) = 36x² + 48x + 15

Therefore, g(f(x)) = 36x² + 48x + 15

Begin with the innermost function: Find f(3) by substituting 3 for x in the function f(x): f(3) = 2(3) + 1 = 6 + 1 = 7. Next, substitute the result of that function for x in g(x); g(7) = 7 – 2 = 5. Finally, substitute 5 for x in f(x): f(5) = 2(5) + 1 = 10 + 1 = 11.

Begin with the innermost function: Find f(–2) by substituting –2 for x in the function f(x): f(–2) = 3(–2) + 2 = –6 + 2 = –4. Then, substitute the result of that function for x in g(x); g(–4) = 2(–4) – 3 = –8 – 3 = –11.

Begin with the innermost function: find g(–3) by substituting –3 for x in the function g(x):g(–3) = (–3)² = 9. Then, substitute the result of that function for x in f(x); f(9) = 2(9) – 1 = 18 – 1 = 17.

Draw a horizontal line across the coordinate plane where f(x) = –2. This line does not touch the graph of f(x) at all. Therefore, there are 0 values for which f(x) = –2.

Draw a horizontal line across the coordinate plane where f(x) = –5. This line touches the graph of f(x) in 8 places. Therefore, there are 8 values for which f(x) = –5.

f(x) = 0 every time the graph touches the x-axis, since the x-axis is the graph of the line f(x) = 0. The graph of f(x) touches the x-axis in 5 places. Therefore, there are 5 values for which f(x) = 0.

Draw a horizontal line across the coordinate plane where f(x) = 3. This line touches the graph of f(x) in 1 place. Therefore, there is 1 value for which f(x) = 3.

Draw a horizontal line across the coordinate plane where f(x) = 2. In other words, graph the line y = 2. This line touches the graph of f(x) in 3 places. Therefore, there are 3 values for which f(x) = 2.