Functions：練習問題

Given the functions \(f(x) = 2x - 1\) and \(g(x) = x^2 + 3\), find the value of the composite function \(gf(2)\).

Find the inverse function, \(f^{-1}(x)\), for \(f(x) = x + 4\).

Find the coordinates of the vertex of the quadratic function \(f(x) = x^2 - 6x + 5\).

A quadratic function \(f(x)\) has a vertex at \((3, -1)\) and passes through the point \((1, 7)\). Find the value of the stretch factor \(a\) when the function is written in the form \(f(x) = a(x - h)^2 + k\).

Consider the functions \( f(x) = 4x - 5 \) and \( g(x) = 3(x + 2) \).

(a) Calculate the value of \( f(6) \).
(b) Find the value of \( x \) when \( g(x) = 27 \).
(c) Work out the value of \( f(2) + g(1) \).

Consider the functions \( f(x) = \frac{2x - 3}{5} \) and \( g(x) = x^2 + 1 \).

(a) Calculate the value of \( f(9) \).
(b) Find the inverse function \( f^{-1}(x) \).
(c) Find the composite function \( g(f(x)) \). Give your answer in the form \( \frac{ax^2 + bx + c}{25} \).