Specialist2025 Exam 1 · Q9Tech-free6 marks

Rational function with a removable discontinuity

Let $f: \R \setminus \{-1, 1\} \to \R,\ f(x) = \dfrac{x^3 + x^2 - 2x}{1 - x^2}$ .

Show that $f(x)$ can be written in the form $f(x) = -x - 1 + \dfrac{1}{x+1}$ , for $x \in \R \setminus \{-1, 1\}$ .

2 marks

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Consider the function with rule

$g(x) = \begin{cases} \dfrac{x^3 + x^2 - 2x}{1 - x^2}, & x \in \R \setminus \{-1, 1\} \\ k, & x \in \{1\} \end{cases}$

Find the value of $k$ such that the graph of $g$ is continuous at $x = 1$ .

1 mark

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Sketch the graph of $y = f(x)$ on the axes below.

Label the asymptotes with their equations.

Blank Cartesian axes on a grid, with x and y each marked from −4 to 4 in steps of 2, provided for sketching the graph of y = f(x).

3 marks

Hints, answer checking and worked solutions come with a Season Pass.