Functions：練習問題

The functions \(f\) and \(g\) are defined by \(f(x) = 2x + 3\) and \(g(x) = 5x - 1\) for all real \(x\). Find the value of \(fg(2)\).

The function \(f\) is defined by \(f(x) = \sqrt{2x + 5}\) for \(x \ge -2.5\). Find an expression for the inverse function \(f^{-1}(x)\).

A function \(f\) is defined by \(f(x) = \frac{ax+b}{x+c}\) for \(x \neq -c\). Given that the function is its own inverse such that \(f(x) = f^{-1}(x)\) for all \(x\) in the domain, which of the following conditions must be satisfied?

A function \( f \) is defined by \( f(x) = e^{2x} - 3 \) for all real \( x \). Find an expression for the inverse function \( f^{-1}(x) \).

The function \(f\) is defined by \(f(x) = \frac{k}{x-2}\) for \(x \neq 2\). Given that the value of the constant \(k\) is such that \(f^2(4) = 4\), find \(k\).

The function \(f\) is defined by \(f(x) = 4\mathrm{e}^{2x} - 3\) for \(x \in \mathbb{R}\).
Find an expression for the inverse function \(f^{-1}(x)\) and state its domain.

The functions \(f\) and \(g\) are defined by
\(f(x) = \mathrm{e}^{2x} - 4\) for \(x \in \mathbb{R}\),
\(g(x) = \ln(3x + 1)\) for \(x > -\frac{1}{3}\).
Solve the equation \(fg(x) = 21\), giving your answer in exact form.

A function \(h\) is defined by \(h(x) = (x-3)^2 + 1\) for \(x \in \mathbb{R}\). State the smallest value of \(k\) such that \(h\) has an inverse when its domain is restricted to \(x \ge k\). Hence, find the expression for \(h^{-1}(x)\) for this restricted domain.

A function \( g \) is defined by \( g(x) = \frac{x+1}{2} \) for \( x \in \mathbb{R} \).

(a) Explain why \( g \) is a one-one function.

(b) Find an expression for \( g^{-1}(x) \).

The function h is defined by \(h(x) = |x^2 - 4x + 3|\) for \(x \in \mathbb{R}\).

(a) Write down the coordinates of the turning point of the graph \(y = x^2 - 4x + 3\).

(b) Hence, find the range of \(h(x)\).

(c) State the values of \(x\) for which \(h(x) = 0\).