Question

Find the number of distinct real solutions of the equation f(f(f(x))) = 0, where f(x) = x² – 1 

Answer 1

We have f(f(f(x))) = 0

⇒ f(f(x² – 1)) = 0

⇒ f ((x² – 1)² – 1) = 0

⇒ f((x⁴ – 2x² + 1 – 1)) = 0

⇒ f(x⁴ – 2x²) = 0

⇒ (x⁴ – 2x²)² – 1 = 0

⇒ (x⁴ – 2x²)² = 1

⇒ (x⁴ – 2x²) = ± 1

When (x⁴ – 2x²) = – 1 ⇒ x = ± 1

From the graph it is clear that, it has two solutions Thus, the number of distinct real solutions of the given equation is 4.