The question presents the following function definition:

$f: x \to \frac{x + b}{x + a}, \quad x \in \mathbb{R}, x \neq -a$

And it asks for the following:

a) State the value of the constant $a$.

• We are given that $f(6) = 4$.

b) Find the value of the constant $b$.

c) Find the inverse of $f$, denoted as $f^{-1}(x)$, and state its domain.

Part a: Solving for $a$

We are given $f(6) = 4$. Using the function formula:

$f(x) = \frac{x + b}{x + a}$

Substituting $x = 6$ and $f(6) = 4$:

$\frac{6 + b}{6 + a} = 4$

Multiplying both sides by $6 + a$:

$6 + b = 4(6 + a)$

Expanding the right-hand side:

$6 + b = 24 + 4a$

Solving for $b$:

$b = 24 + 4a - 6 = 18 + 4a$

Part b: Finding the value of $b$

At this point, we still need an additional condition to solve for $a$, but we can express $b$ in terms of $a$ as $b = 18 + 4a$.

Next, would you like more details on solving for $a$ and $b$, or shall we move on to finding the inverse function in part c?