Calculate \(f(1)\).

Sketch the graph of \(y = f(x)\) for \( - 7 \leqslant x \leqslant 4\) and \( - 30 \leqslant y \leqslant 30\).

Write down the equation of the vertical asymptote.

Write down the coordinates of the \(x\)-intercept.

Write down the possible values of \(x\) for which \(x < 0\) and \(f’(x) > 0\).

Find the solution of \(f(x) = g(x)\).

\(0.3{(1)^3} + \frac{{10}}{1} + {2^{ - 1}}\)     (M1)

Note:     Award (M1) for correct substitution into function.

\( = 10.8\)     (A1)(G2)

[2 marks]

     (A1)(A1)(A1)(A1)

Note:     Award (A1) for indication of correct window and labelled axes.

Award (A1) for correct shape and position for \(x < 0\) (with the local maximum, local minimum and \(x\)-intercept in relative approximate location in \({{\text{3}}^{{\text{rd}}}}\) quadrant).

Award (A1) for correct shape and position for \(x > 0\) (with the local minimum in relative approximate location in \({{\text{1}}^{{\text{st}}}}\) quadrant).

Award (A1) for smooth curve with indication of asymptote (graph should not touch \(y\)-axis and should not curve away from the \(y\)-axis). The asymptote is only assessed in this mark.

[4 marks]

\(x = 0\)     (A2)

Note:     Award (A1) for “\(x = {\text{(a constant)}}\)” and (A1) for “\({\text{(a constant)}} = 0\)”.

The answer must be an equation.

[2 marks]

\(( - 6.18,{\text{ }}0){\text{ }}( - 6.17516 \ldots ,{\text{ }}0)\)     (A1)(A1)

Note:     Award (A1) for each correct coordinate. Award (A0)(A1) if parentheses are missing.

[2 marks]

\( - 4.99 < x <  - 2.47{\text{ }}( - 4.98688 \ldots  < x <  - 2.46635 \ldots )\)     (A1)(A1)

Note:     Award (A1) for both correct end points, (A1) for strict inequalities used with 2 endpoints.

[2 marks]

\(0.3{x^3} + \frac{{10}}{x} + {2^{ - x}} = 2x - 3\)     (M1)

Note:     Award (M1) for equating the expressions for \(f\) and \(g\) or for the line \(y = 2x - 3\) sketched (positive gradient, negative \(y\)-intercept) on their graph from part (a).

\((x = ){\text{ }} - 1.34{\text{ }}( - 1.33650 \ldots )\)     (A1)(G2)

Note:     Award a maximum of (M1)(A0) or (G1) for coordinate pair seen as final answer.

[2 marks]