The function \(f(x) = 1.1\) xrepresents the score increased by 10%

The function \(g(x) = x + 8\) represents the score increased by 8 points

The function \((f*g)(x) = 1.1(x + 8)\) represents the final score when the score is first increased by 8 bonus points and then by 10%

The function \((g*f)(x) = l.lx + 8\) represents the final score when the score is first increased by 10% and then by 8 bonus points

A score of 90 or better results in an A

Calculation:

(i) Consider \((f*g)(x) = 1.1 (x+8)\)

Plugging the final score of 90,

\(90 = 1.1 (x+8)\)

Dividing by 1.1 on both the sides,

\(\frac{90}{1.1}=\frac{1.1(x+8)}{1.1}\)

\(81.8181... =x+8\)

\(x+8=818181...\)

\(x+8= 81.82\)

Subtracting 8 from both the sides,

\(x+8-8=81.82-8\)

\(x= 73.82\)

(ii) Consider \((g*f)(x) = 1.1x +8\)

Plugging the final score of 90,

\(90 = 1.1x+8\)

Subtracting 8 from both the sides,

\(90-8 = 1.1x+8-8\)

\(82= 1.1x\)

\(1.1x = 82\)

Dividing by 1.1 on both the sides,

\(\frac{1.1x}{1.1}=\frac{82}{1.1}\)

\(x= 74.5454...\)

\(x= 74.55\).

The function \(g(x) = x + 8\) represents the score increased by 8 points

The function \((f*g)(x) = 1.1(x + 8)\) represents the final score when the score is first increased by 8 bonus points and then by 10%

The function \((g*f)(x) = l.lx + 8\) represents the final score when the score is first increased by 10% and then by 8 bonus points

A score of 90 or better results in an A

Calculation:

(i) Consider \((f*g)(x) = 1.1 (x+8)\)

Plugging the final score of 90,

\(90 = 1.1 (x+8)\)

Dividing by 1.1 on both the sides,

\(\frac{90}{1.1}=\frac{1.1(x+8)}{1.1}\)

\(81.8181... =x+8\)

\(x+8=818181...\)

\(x+8= 81.82\)

Subtracting 8 from both the sides,

\(x+8-8=81.82-8\)

\(x= 73.82\)

(ii) Consider \((g*f)(x) = 1.1x +8\)

Plugging the final score of 90,

\(90 = 1.1x+8\)

Subtracting 8 from both the sides,

\(90-8 = 1.1x+8-8\)

\(82= 1.1x\)

\(1.1x = 82\)

Dividing by 1.1 on both the sides,

\(\frac{1.1x}{1.1}=\frac{82}{1.1}\)

\(x= 74.5454...\)

\(x= 74.55\).