Final answer:
The value of m that makes G(x) a self-inverse function is m = -2. This result is obtained by setting G(G(x)) equal to x and solving the resulting equations through algebraic manipulations.
Step-by-step explanation:
To find the value of m such that G(x) is a self-inverse function, we must set G(G(x)) = x. Let's start by setting G(x) equal to y, resulting in the equation y = (2x - 4) / (x + m). Solving for x in terms of y gives us x = (4 + my) / (2 - y). To ensure the function is self-inverse, we substitute x with G(x) in this equation and solve for y to get the original input x.
Thus, x = (4 + mG(x)) / (2 - G(x)), replacing G(x) with y we have x = (4 + my) / (2 - y). Conducting the substitution, we get x = (4 + m((2x - 4) / (x + m))) / (2 - ((2x - 4) / (x + m))). Solving for x will give us an equation with x on both sides—if we can simplify this to just x = x, we know we've found the correct m.
The simplification process involves algebraic manipulations. After performing the necessary calculations (which include multiplying both sides by the denominators to eliminate the fractions, collecting like terms, and solving for m), we find that the value of m that makes G(x) a self-inverse function is m = -2.