Asymptotes：練習問題

A function is given by \(f(x) = \frac{2x - 5}{x + 3}\). What are the equations of the vertical and horizontal asymptotes for the graph of this function?

Consider the function \(f(x) = \frac{2x + 1}{x - 5}\). Identify the equations of the vertical asymptote and the horizontal asymptote for the graph of \(y = f(x)\).

For the trigonometric function \(g(x) = \tan(2x)\), which of the following is an equation of an asymptote in the interval \(0^\circ \le x \le 180^\circ\)?

Identify the equation of the vertical asymptote for the function \(f(x) = \frac{3}{2x - 5} + 4\).

A function is defined as \(h(x) = \frac{ax + 5}{x - b}\). The graph of \(y = h(x)\) has a vertical asymptote at \(x = -4\) and a horizontal asymptote at \(y = 3\). Calculate the value of \(a + b\).

State the equation of the horizontal asymptote for the graph of the function \(f(x) = \frac{1}{x} + 7\).

Consider the function \( f(x) = \frac{3x^2 - 12}{x^2 - x - 6} \). By simplifying the expression, identify the equation of the vertical asymptote of the graph of \( y = f(x) \).

Consider the function \(g(x) = \frac{1}{x+2}\).
(a) State the equations of the vertical and horizontal asymptotes for the graph of \(y = g(x)\).
(b) The graph of \(y = g(x)\) is translated by the vector \(\begin{pmatrix} -1 \\ 3 \end{pmatrix}\) to form the graph of \(y = h(x)\). Write down the equation of \(h(x)\) and state the equations of its new vertical and horizontal asymptotes.

Consider the function
f(x) = \(\frac{2}{x-3} + 1\)

(a) Write down the equation of the vertical asymptote of the graph of y = f(x).

(b) Write down the equation of the horizontal asymptote of the graph of y = f(x).

(c) Explain briefly what an asymptote represents in relation to the graph of a function.