Question

If \( f(x)=\dfrac{4x+7}{7x-4} \), then the value of \[ f\{f[f(2)]\} \] is

• A. \( \dfrac{3}{2} \)
• B. \( \dfrac{2}{3} \)
• C. \( \dfrac{35}{39} \)
• D. \( \dfrac{39}{35} \)

Hint:

In composite functions, always evaluate step by step from the innermost function.

Answer 1

The Correct Option is A

Step 1: Find \(f(2)\).
\[ f(2)=\frac{4(2)+7}{7(2)-4}=\frac{8+7}{14-4}=\frac{15}{10}=\frac{3}{2} \]
Step 2: Find \(f[f(2)] = f\!\left(\dfrac{3}{2}\right)\).
\[ f\!\left(\frac{3}{2}\right) =\frac{4\left(\frac{3}{2}\right)+7}{7\left(\frac{3}{2}\right)-4} =\frac{6+7}{\frac{21}{2}-4} =\frac{13}{\frac{13}{2}}=2 \]
Step 3: Find \(f\{f[f(2)]\ = f(2)\).}
\[ f(2)=\frac{3}{2} \]
Step 4: Final Answer.
\[ f\{f[f(2)]\}=\frac{3}{2} \]