(a) If \(f(x) = \frac{4 – 5x}{2}\), and \(g(x) = x + 6, x \in R\), find \(f \circ g^{-1}\).
(b) P(x, y) divides the line joining (7, -5) and (-2, 7) internally in 5 : 4. Find the coordinates of P.
Explanation
(a) \(f(x) = \frac{4 - 5x}{2} ; g(x) = x + 6, x \in R\)
Let g(x) = y
\(y = x + 6 \implies x = y - 6\)
Let \(x = g^{-1} (x)\) and y = x.
\(g^{-1}(x) = x - 6\)
\(f \cdot g^{-1}(x) = f(g^{-1}(x))\)
= \(\frac{4 - 5(x - 6)}{2}\)
= \(\frac{4 - 5x + 30}{2}\)
= \(\frac{34 - 5x}{2}\)
(b) \(x = \frac{7 \times 4 - 2(5)}{5 + 4}\)
= \(\frac{18}{9}\)
\(x = 2\)
\(y = \frac{4(-5) + 5(7)}{5 + 4}\)
= \(\frac{15}{9}\)
\((x, y) = (2, 1\frac{2}{3})\)