If a function \(f : Q \to R\) is defined by \(f(x)=\displaystyle\frac{2x-1}{2}\) and \(g : Q \to R\) is defined by \(g(x)= x+2\), then \((g\circ f)\left(\displaystyle\frac{3}{2}\right)\) is :
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 3
By the definition of composition, \((g\circ f)(x) = g(f(x))\): apply f first, then g.
\( \displaystyle f\left(\frac{3}{2}\right) = \frac{2\cdot\frac{3}{2} - 1}{2} = \frac{3 - 1}{2} = 1 \)
\( \displaystyle (g\circ f)\left(\frac{3}{2}\right) = g(1) = 1 + 2 = 3 \)
Why the other options are incorrect
Option (1) 1 — Incorrect. 1 is only \(f\left(\frac{3}{2}\right)\); g has not been applied yet.
Option (2) 2 — Incorrect. \((g\circ f)\left(\frac{3}{2}\right) = g(1) = 3\), not 2.
Option (3) \(\displaystyle\frac{7}{2}\) — Incorrect. This is \(g\left(\frac{3}{2}\right) = \frac{3}{2} + 2\), which leaves out f.