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Question
Let \(f(x) = m – \frac{1}{x}\), for \(x \ne 0\). The line \(y = x – m\) intersects the graph of \(f\) in two distinct points. Find the possible values of \(m\).
Answer/Explanation
Markscheme
valid approach (M1)
eg\(\,\,\,\,\,\)\(f = y,{\text{ }}m – \frac{1}{x} = x – m\)
correct working to eliminate denominator (A1)
eg\(\,\,\,\,\,\)\(mx – 1 = x(x – m),{\text{ }}mx – 1 = {x^2} – mx\)
correct quadratic equal to zero A1
eg\(\,\,\,\,\,\)\({x^2} – 2mx + 1 = 0\)
correct reasoning R1
eg\(\,\,\,\,\,\)for two solutions, \({b^2} – 4ac > 0\)
correct substitution into the discriminant formula (A1)
eg\(\,\,\,\,\,\)\({( – 2m)^2} – 4\)
correct working (A1)
eg\(\,\,\,\,\,\)\(4{m^2} > 4,{\text{ }}{m^2} = 1\), sketch of positive parabola on the \(x\)-axis
correct interval A1 N4
eg\(\,\,\,\,\,\)\(\left| m \right| > 1,{\text{ }}m < – 1\) or \(m > 1\)
[7 marks]