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]