(a)

(i) \( x = -4 \) (A1, N1).

Working:
The vertical asymptote occurs where the denominator is zero: \( x + 4 = 0 \), so \( x = -4 \). (A1)

(ii) \( y = 4 \) (M1A1, N2).

Working:
For the horizontal asymptote, evaluate the limit as \( x \to \infty \):
\( f(x) = \frac{4x + 1}{x + 4} \).
\( \lim_{x \to \infty} \frac{4x + 1}{x + 4} = \lim_{x \to \infty} \frac{4 + \frac{1}{x}}{1 + \frac{4}{x}} = \frac{4 + 0}{1 + 0} = 4 \).
Similarly, as \( x \to -\infty \): \( \lim_{x \to -\infty} \frac{4x + 1}{x + 4} = 4 \).
Thus, the horizontal asymptote is \( y = 4 \). (M1 for limit attempt, A1 for answer)

[3 marks]

(b)

(i) \( f^{-1}(x) = \frac{1 – 4x}{x – 4} \) (M1A1A1A1, N3).

Working:
To find the inverse, set \( y = f(x) \): \( y = \frac{4x + 1}{x + 4} \).
Solve for \( x \): \( y (x + 4) = 4x + 1 \). (M1)
\( xy + 4y = 4x + 1 \). (A1)
\( xy – 4x = 1 – 4y \). (A1)
\( x (y – 4) = 1 – 4y \).
\( x = \frac{1 – 4y}{y – 4} \).
Thus, \( f^{-1}(x) = \frac{1 – 4x}{x – 4} \). (A1 for final answer)

(ii) (M1A1M1A1A1, N3).

Working:
Reflection in y-axis given by \( f(-x) \): (M1)
\( f(-x) = \frac{4(-x) + 1}{-x + 4} = \frac{-4x + 1}{-x + 4} \). (A1)
Reflection of \( f(-x) \) in x-axis given by \( -f(-x) \): (M1)
\( -f(-x) = -\left( \frac{-4x + 1}{-x + 4} \right) = \frac{4x – 1}{-x + 4} \). (A1)
Simplify: \( \frac{4x – 1}{-x + 4} = \frac{-(1 – 4x)}{-(x – 4)} = \frac{1 – 4x}{x – 4} \). (A1)
\( = f^{-1}(x) \). (AG)

[6 marks]

(c)

(i) \( p = -1 \), \( q = 1 \) (M1A1, N2).

Working:
Find intersection points by setting \( f(x) = f^{-1}(x) \):
\( \frac{4x + 1}{x + 4} = \frac{1 – 4x}{x – 4} \). (M1)
Cross-multiply: \( (4x + 1)(x – 4) = (1 – 4x)(x + 4) \).
\( 4x^2 – 16x + x – 4 = x + 4 – 4x^2 – 16x \).
\( 4x^2 – 15x – 4 = -4x^2 – 15x + 4 \).
\( 8x^2 – 8 = 0 \).
\( x^2 = 1 \).
\( x = \pm 1 \).
Since \( p < q \), \( p = -1 \), \( q = 1 \). (A1)

(ii) Area \( \approx 0.675 \) units² (M1A1A1A1, N4).

Working:
attempt to set up an integral to find area between \( f \) and \( f^{-1} \) (M1)
\( \int_{-1}^{1} \left( \frac{4x + 1}{x + 4} – \frac{1 – 4x}{x – 4} \right) dx \) (A1)
= 0.675231… (A1)
= 0.675 (A1)

[4 marks]

Total [16 marks]