IB DP Maths Topic 2.5 The reciprocal function x↦1x , x≠0 : its graph and self-inverse nature SL Paper 2

 

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Question

Let \(f\left( x \right) = \frac{{8x – 5}}{{cx + 6}}\) for \(x \ne  – \frac{6}{c},\,\,c \ne 0\).

The line x = 3 is a vertical asymptote to the graph of f. Find the value of c.

[2]
a.

Write down the equation of the horizontal asymptote to the graph of f.

[2]
b.

The line y = k, where \(k \in \mathbb{R}\) intersects the graph of \(\left| {f\left( x \right)} \right|\) at exactly one point. Find the possible values of k.

[3]
c.
Answer/Explanation

Markscheme

valid approach       (M1)
eg  \(cx + 6 = 0,\,\, – \frac{6}{c} = 3\)

c = −2      A1 N2

[2 marks]

a.

valid approach (M1)
eg  \(\mathop {{\text{lim}}\,f}\limits_{x \to \infty } \left( x \right),\,\,y = \frac{8}{c}\)

y = −4 (must be an equation)      A1 N2

[2 marks]

b.

valid approach to analyze modulus function      (M1)
eg   sketch, horizontal asymptote at y = 4, y = 0

k = 4, k = 0     A2 N3

[3 marks]

c.
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