(a) Composition of functions:

Solution:

To find \( (g \circ f)(x) \), we substitute \( f(x) \) into \( g(x) \):

\[ (g \circ f)(x) = g(f(x)) = (x – 3)^2 + k^2 \]

Expanding the expression:

\[ (g \circ f)(x) = x^2 – 6x + 9 + k^2 \]

Thus, the final expression is:

\[ (g \circ f)(x) = x^2 – 6x + (9 + k^2) \]

[2 marks]

(b) Finding possible values of k:

Solution:

Given \( (g \circ f)(2) = 10 \), we substitute \( x = 2 \) into our expression:

\[ (2 – 3)^2 + k^2 = 10 \]

\[ 1 + k^2 = 10 \]

\[ k^2 = 9 \]

Taking square roots:

\[ k = \pm 3 \]

Therefore, the possible values of \( k \) are \( 3 \) and \( -3 \).

[3 marks]