Functions：練習問題

A function \( f \) is defined by \( f(x) = \frac{1}{x-2} \) for the domain \( D_f = (2, 5] \). Find the range of \( f \).

A function \( f \) is defined by \( f(x) = \frac{1}{x^2 + 1} \) for the domain \( x \in \mathbb{R}, x \leq 0 \).
Given that the function \( g \) is defined as \( g: x \mapsto 2 - e^x \) for \( x \in \mathbb{R}, x \leq a \), determine the largest value of \( a \) such that the composite function \( fg \) exists.

The function \( f \) is defined by \( f: x \mapsto \frac{2x+3}{x-1} \) for \( x \in \mathbb{R}, x \neq 1 \).
The function \( g \) is defined by \( g: x \mapsto x^2 - 4x + 1 \) for \( x \in \mathbb{R}, x \geq k \).
If the composite function \( f g \) exists and \( g \) has an inverse, find the smallest possible value of the constant \( k \).

Let the function \( f \) be defined by \( f: x \mapsto \ln(x-1) \) for \( x \in \mathbb{R}, x > 1 \) and the function \( g \) be defined by \( g: x \mapsto e^x + k \) for \( x \in \mathbb{R} \), where \( k \) is a constant. Find the set of values of \( k \) such that the composite function \( fg \) exists.

The function \( f \) is defined by \( f(x) = \frac{x+2}{x-1} \) for \( x \in \mathbb{R}, x \neq 1 \). The function \( g \) is defined by \( g(x) = x^2 - 4x + c \) for \( x \in \mathbb{R}, x \le 2 \), where \( c \) is a constant. Find the set of values of \( c \) such that the composite function \( fg \) exists.

The function \( f \) is defined by \( f(x) = \frac{1}{x-2} \) for \( x \in \mathbb{R}, x > 2 \). State the range of the function \( f \).

The function \( f \) is defined by \( f(x) = \frac{4}{2-x} \) for \( x < 2 \). The function \( g \) is defined by \( g(x) = x^2 + 1 \) for \( x \in \mathbb{R} \). Determine, with a reason, whether the composite function \( fg \) exists, and find the range of \( f \).

The functions \( f \) and \( g \) are defined as follows:
\( f: x \mapsto \ln(x - 1), \quad x \in \mathbb{R}, x > 1 \)
\( g: x \mapsto x^2 + k, \quad x \in \mathbb{R} \)
Find the range of values of the constant \( k \) for which the composite function \( fg \) exists.

The function \( f \) is defined by \( f(x) = \frac{4x+1}{x-2} \) for \( x \in \mathbb{R}, x \neq 2 \).
(a) Show that \( f^{-1}(x) \) exists and find an expression for \( f^{-1}(x) \).
(b) State the domain and range of \( f^{-1} \).
(c) Sketch the graphs of \( y = f(x) \) and \( y = f^{-1}(x) \) on the same diagram, showing clearly the equations of any asymptotes and the coordinates of any points of intersection with the axes.

A function \( f \) is defined by \( f(x) = 3 + \sin\left(\frac{\pi x}{2}\right) \) for the domain \( 1 \le x \le 3 \).
(i) Show that \( f \) has an inverse and find an expression for \( f^{-1}(x) \).
(ii) Sketch the graph of \( f \) and use it to find the range of \( f \).
(iii) Solve the equation \( f(x) = f^{-1}(x) \).