Exam-Style Questions.

(a) Sketch the graph of \( y = f(x) \) for values of \( x \) between \(-5\) and \(5\) given that:

$$ f(x) = \frac{1}{x-1} - x $$

(b) Write down the equations of any asymptotes parallel to the \( y \)-axis.

(c) Find the coordinates of the y-axis intercept.

(d) Find the coordinates of the x-axis intercepts.

(e) Find a value of \( k \) for which \( g(x) = kx \) does not intercept \( f(x)\).

(f) Draw the graph of \(h(x)=x-3\) on the same set of axes.

(g) Solve the equation \( f(x) = h(x) \).

(h) Find the solutions to the inequality \( f(x) > h(x) \).

Worked Solution

The function \( f \) is defined by \( f(x) = \frac{5x + 5}{3x - 6} \) for \( x \in \mathbb{R}, x \neq 2 \).

(a) Find the zero of \( f(x) \).

(b)For the graph of \( y = f(x) \), write down the equation of the asymptotes;

(c) Find \( f^{-1}(x) \), the inverse function of \( f(x) \).

Worked Solution

A function \(f\) is defined by \(f(x) = 2 + \dfrac{1}{3-x}, \text{ where } x \in \mathbb{R}, x \neq 3.\)

The graph of \(y=f(x)\) has a vertical asymptote and a horizontal asymptote.

(a) Write down the equation of the horizontal asymptote;

(b) Write down the equation of the vertical asymptote;

Find the coordinates of the point where the graph of \(y\) intersects:

(c) the y-axis;

(d) the x-axis.

Worked Solution

Let \(f(x)=\frac{3x}{x-q}\), where \(x \neq q\).

(a) Write down the equations of the vertical and horizontal asymptotes of the graph of \(f\).

The vertical and horizontal asymptotes to the graph of \(f\) intersect at the point Q(1, 3).

(b) Find the value of q.

(c) The point P(x, y) lies on the graph of \(f\). Show that PQ = \(\sqrt{(x-1)^2+(\frac{3}{x-1})^2}\)

(d) Hence find the coordinates of the points on the graph of \(f\) that are closest to (1, 3).

Worked Solution

Let \(f(x) = \frac{9x-3}{bx+9}\) for \(x \neq -\frac9b, b \neq 0\).

(a) The line \(x = 3\) is a vertical asymptote to the graph of \(f\). Find the value of b.

(b) Write down the equation of the horizontal asymptote to the graph of \(f\).

(c) The line \(y = c\) , where \(c\in \mathbb R\) intersects the graph of \( \begin{vmatrix}f(x) \end{vmatrix} \) at exactly one point. Find the possible values of \(c\).

Worked Solution