Functions：練習問題

The function \(f\) is defined by \(f(x) = 3x - 2\) for the domain \(1 \leq x \leq 5\). Find the range of \(f\).

The function \(f\) is defined by \(f(x) = \frac{2x+1}{x-3}\) for \(x \neq 3\). Find an expression for the inverse function \(f^{-1}(x)\).

The functions \( f \) and \( g \) are defined by \( f(x) = x^2 + 2x \) for \( x \in \mathbb{R} \) and \( g(x) = \frac{1}{x + 4} \) for \( x > -4 \). Find the range of the composite function \( gf \).

The function \( f \) is defined by \( f(x) = \sqrt{x - 3} \) for the domain \( x \ge 3 \). State the range of \( f \).

The function \( f \) is defined by \( f(x) = (x-2)^2 + 3 \) for \( x \ge 2 \). Find an expression for \( f^{-1}(x) \).

Given the function \( f(x) = 2x + 5 \), find the value of the composite function \( ff(1) \).

The function \( f \) is defined by \( f(x) = ax + b \) for \( x \in \mathbb{R} \), where \( a \) and \( b \) are constants and \( a > 0 \). Given that \( f(f(x)) = 9x - 8 \), find the values of \( a \) and \( b \).

The function \( f \) is defined by \( f(x) = \sqrt{x+4} \) for \( x \ge -4 \). The graph of \( y = g(x) \) is obtained by reflecting the graph of \( y = f(x) \) in the line \( y = x \) and then translating the resulting graph by \( 3 \) units in the positive \( y \)-direction. Find an expression for \( g(x) \) and state its domain.

The functions \( f \) and \( g \) are defined for \( x \in \mathbb{R} \) by:
\( f(x) = 3x - 1 \)
\( g(x) = \frac{2}{x-2} \) for \( x \neq 2 \)

(a) Find an expression for the composite function \( fg(x) \).
(b) Solve the equation \( fg(x) = 5 \).

The functions \( f \) and \( g \) are defined by:
\( f(x) = \frac{6}{x-3} \) for \( x > 3 \)
\( g(x) = 2x + 1 \) for \( x > 1 \)

(a) Find an expression for the composite function \( fg(x) \).
(b) Find an expression for the inverse function \( (fg)^{-1}(x) \) and state its domain.
(c) State the range of \( fg \).