Question

If \( f(x) = ax + b \), \(a\) and \(b\) are positive real numbers and if \( f(f(x)) = 9x + 8 \), then the value of \( a + b \) is:

• A. 3
• B. 4
• C. 5
• D. 6
• E. None of the above

Hint:

When solving composite function problems like \(f(f(x))\), always expand systematically and then compare coefficients of like terms to solve for unknown constants.

Answer 1

The Correct Option is C

Step 1: Write the given function
We are given: \[ f(x) = ax + b \]

Step 2: Apply the function twice
Now, \[ f(f(x)) = f(ax + b) \] Substitute into the function: \[ f(ax + b) = a(ax + b) + b = a^2x + ab + b \]

Step 3: Compare with the given condition
We are told: \[ f(f(x)) = 9x + 8 \] So, \[ a^2x + ab + b = 9x + 8 \]

Step 4: Equating coefficients
From coefficients of \(x\): \[ a^2 = 9 \quad \Rightarrow \quad a = 3 \quad (\text{since } a>0) \] From constant term: \[ ab + b = 8 \quad \Rightarrow \quad b(a + 1) = 8 \] Substitute \(a = 3\): \[ b(3 + 1) = 8 \quad \Rightarrow \quad 4b = 8 \quad \Rightarrow \quad b = 2 \]

Step 5: Find \(a + b\)
\[ a + b = 3 + 2 = 5 \] \[ \boxed{5} \]