Monotonic Function
A monotonic function is a function that is either entirely increasing or entirely decreasing. In other words, if we have two values of x, say x1 and x2, such that x1 < x2,="" then="" the="" value="" of="" f(x1)="" will="" be="" less="" than="" or="" equal="" to="" f(x2)="" if="" the="" function="" is="" increasing,="" or="" greater="" than="" or="" equal="" to="" f(x2)="" if="" the="" function="" is="" />

f(x) = f^(-1)(x)
This equation states that the function f(x) is equal to its inverse function f^(-1)(x). In other words, if we have a value x and we apply the function f(x) to it, and then apply the inverse function f^(-1)(x) to the result, we should get back to the original value of x.

Finding the Value of a
To find the value of a that satisfies the given conditions, let's substitute f(x) = ax + b into the equation f(x) = f^(-1)(x). We get:

ax + b = f^(-1)(x)

Now, let's find the inverse of f(x). To do this, we can interchange x and y in the equation and solve for y:

y = ax + b
x = ay + b
x - b = ay
y = (x - b)/a

The inverse function is therefore f^(-1)(x) = (x - b)/a. Substituting this into the equation, we get:

ax + b = (x - b)/a

To simplify further, let's multiply both sides of the equation by a:

a^2x + ab = x - b

Now, let's rearrange the equation to isolate the x term:

(a^2 - 1)x = -ab - b

Finally, divide both sides by (a^2 - 1) to solve for x:

x = (-ab - b)/(a^2 - 1)

Since this equation should hold true for all values of x, the numerator and denominator must be equal to zero:

-ab - b = 0 (equation 1)
a^2 - 1 = 0 (equation 2)

Solving equation 1 for b, we get:

b = -ab

Substituting this into equation 2, we get:

a^2 - 1 = 0
a^2 = 1
a = ±1

Therefore, the possible values for a are 1 and -1. However, since the question specifies that f(x) is a monotonic function, we can conclude that a = -1, as an increasing function cannot have a = 1.

Hence, the value of a that satisfies the given conditions is -1.