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Exam-Style Question on Graphs

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Question id: 711. This question is similar to one that appeared on a GCSE Higher paper in 2022. The use of a calculator is allowed.

(a) Sketch the graph of \( y = f(x) \) for values of \( x \) between \(-5\) and \(5\) given that:

$$ f(x) = \frac{1}{x-1} - x $$

(b) Write down the equations of any asymptotes parallel to the \( y \)-axis.

(c) Find the coordinates of the y-axis intercept.

(d) Find the coordinates of the x-axis intercepts.

(e) Find a value of \( k \) for which \( g(x) = kx \) does not intercept \( f(x)\).

(f) Draw the graph of \(h(x)=x-3\) on the same set of axes.

(g) Solve the equation \( f(x) = h(x) \).

(h) Find the solutions to the inequality \( f(x) > h(x) \).

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