I assume that in the statement of the problem, g(x) = f-1(x) (that is, g(x) is the inverse function for the function f(x).) We are given the linear function f(x) = ax + b, whose graph is a straight line with slope a. Note that as long as a≠0, f(x) is a one-to-one function, and, thus, has an inverse. Let us find the expression for f-1(x). For simplicity, let us first rewrite the function f with y in place of f(x). That is,
y = ax + b
Now solve for x in terms of y.
y - b = ax
Divide both sides by a.
(y - b)/a = x, which may be rewritten as
x = (1/a)y - b/a
To obtain the formula for f-1(x), first interchange x and y in the above, and then replace y with f-1(x).
y = (1/a)x - b/a
f-1(x) = (1/a)x - b/a, and thus
Eq. 1 g(x) = (1/a)x - b/a
We now need to find the constants a and b using the condition that f(x) - g(x) = 44 for all x.
f(x) - g(x) = 44
(ax + b) - ((1/a)x - b/a) = 44
ax + b - (1/a)x + b/a = 44
ax - (1/a)x + b + b/a = 44
Factoring x out of the first two terms gives
Eq. 2 (a - 1/a)x + b + b/a = 44.
The above equation must hold for all values of x. Thus, we can substitute any 2 real numbers we like to obtain a system of two equations which can be solved for a and b. First, set x = 0.
Eq. 3 b + b/a = 44
Now substitute x = 1 in eq. 2 and obtain
Eq. 4 a - 1/a + b + b/a = 44
Note that the last two terms on the right side of eq.4 are b + b/a, and eq.3 says b + b/a = 44. Therefore, we substitute b + b/a = 44 from eq. 3 into eq.4 to obtain
a - 1/a + 44 = 44
a - 1/a = 0
a = 1/a
a2 = 1
a = ±1.
There are 2 possible solutions for a. Substituting a = 1 into eq. 3 and solve for b to obtain
b + b/1 = 44
2b = 44
b = 22
Thus, one solution of the system of equations (eq.3 and eq.4) is a = 1 and b= 22.
Now let a = -1 in eq. 3.
b + b/(-1) = 44
b - b = 44
0 = 44
Thus, if a = -1, the system has no solution. Using a = 1 and b = 22, we now may write f(x) = x + 22. Note also that g(x) = x - 22. You should check that with this choice of f(x) and g(x), the condition f(x) - g(x) = 44 is indeed satisfied. Therefore,
f(x) = x + 22
is the only linear function stisfying the required condition.
Hope that helps! Let me know if you need any clarification.