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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.

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### 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 \( b = 18 + 4a \).

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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?

The function f is defined by f: x → (x + b) / (x + a), x ∈ R, x ≠ -a. State the value of the constant a. Given that f(6) = 4. Find the value of the constant b. Find f^(-1)(x) and state its domain.

Learn how to find the constants a and b, and the inverse of the function f(x) = (x + b) / (x + a) given f(6) = 4. This step-by-step solution covers key algebraic concepts such as solving for unknowns in a rational function and finding its inverse. Suitable for high school students in Grades 11-12.

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