If g(x) = x² + x - 1 and
(goƒ) (x) = 4x² - 10x + 5, then ƒ( 5/4 ) is equal to:

Solution

Maths sir · 4 steps

You do not need f itself: put x = 5/4 into gof, then solve g(t) = that value for t = f(5/4).

1. Name the unknown. Let t = f(5/4). Since (g ° f)(x) = g(f(x)), at x = 5/4 this gives g(t) = (g ° f)(5/4).
2. Evaluate gof. (g ° f)(5/4) = 4 × 25/16 - 10 × 5/4 + 5 = 25/4 - 50/4 + 20/4 = -5/4.
3. Write g(t). With g(t) = t² + t - 1, the condition is t² + t - 1 = -5/4. Move everything to one side and you get a perfect square.

   t² + t + 1/4 = 0 ⇒ (t + 1/2)² = 0

4. So. The only root is t = -1/2, so f(5/4) = -1/2, which is option (c).

The idea to keep. When the question gives g and gof and asks for f at a single point, do not hunt for the full formula of f. Evaluate gof at that point and solve g(t) = that value.

Answer from the NTA final answer key. Solution by PrepWiser.