Considering the steps involved in the first two function machines, we can derive:
f(x) = x^2 + 3x - 4
g(x) = 2x^2 + 6x - 8
= 2(x^2 + 3x - 4)
So g(x) = 2f(x)
This means that the graph of y = g(x) is the same as the graph for y = f(x) but it is stretched in the y axis by a scale factor 2. This is because for whatever value we input for f(x), 2f(x) will always be twice as large; for example if f(x) = 5, then 2f(x) = 10. This also happens for negative y values, as -4 doubled is still -8, for example. However, the two solutions of the quadratic graph y = 2f(x) stay the same because 2 x 0 = 0. Thus, the graph y = g(x) is y = f(x) stretched in the y axis by scale factor 2.
Considering the steps involved in the second two function machines, we can derive:
f(x) = x^2 + 3x - 4
and
g(x) = 4x^2 + 6x - 4
= (2x)^2 + 3(2x) - 4
Wherever we see 'x' in the function f(x), we see '2x' in the function g(x). Therefore, we can write
g(x) = f(2x)
The graph of y = g(x) is the same as the graph of y = f(x) but stretched in the x axis by a scale factor 0.5 (The reciprocal of 2). This is because for any x value 'a' in the graph y = f(x), we take the result of the function of y = f(2x) which would be the result where x = 2a. This is true for negative expressions, as -5 / 2 = -2.5. When x = 0, nothing changes because 0 x 0.5 = 0. So, the graph stretches in the x axis by scale factor 0.5.
The input numbers for which both machines give the same result are the points at which f(x) = g(x). If the graphs of both functions are plotted, it will be the points where they cross (i.e: where a given x value will produce the same y result in each graph). This can be calculated using algebra:
f(x) = x^2 + 3x - 4
g(x) = 4x^2 + 6x - 4
The x value for which g(x) = f(x), therefore, is:
4x^2 + 6x - 4 = x^2 + 3x - 4
3x^2 + 3x = 0 (Subtracting x^2 + 3x - 4 from both sides)
x^2 + x = 0 (Dividing both sides by 3; 0/3 = 0)
x(x + 1) = 0 (Taking out a factor of x)
Therefore the solutions for which x(x + 1) = 0 are 0 and -1:
0(0 + 1) = 0(1) = 0
-1(-1 + 1) = -1(0) = 0
Thus f(x) = g(x) when x = 0 or -1