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 \(m\).

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]