3) The functions $$ f $$ and $$ g $$ are defined by $$ f(x)=6 x+8 $$ and $$ g(x)=\frac{x-2}{3} $$ a) Calculate the value of $$ g\left(\frac{1}{2}\right) $$. b) Write an expression for $$ g f(x) $$ in its simplest form.

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**a)**  
We are given  
$$
g(x)=\frac{x-2}{3}.
$$  
Substitute $$ x=\frac{1}{2} $$ into $$ g(x) $$:  
$$
g\left(\frac{1}{2}\right)=\frac{\frac{1}{2}-2}{3}.
$$  
Simplify the numerator:  
$$
\frac{1}{2}-2=\frac{1}{2}-\frac{4}{2}=\frac{-3}{2}.
$$  
Then,  
$$
g\left(\frac{1}{2}\right)=\frac{-\frac{3}{2}}{3}=-\frac{3}{2}\times\frac{1}{3}=-\frac{1}{2}.
$$

**b)**  
We want to find $$ g(f(x)) $$, where  
$$
f(x)=6x+8 \quad \text{and} \quad g(x)=\frac{x-2}{3}.
$$  
Substitute $$ f(x) $$ into $$ g(x) $$:  
$$
g(f(x))=g(6x+8)=\frac{(6x+8)-2}{3}.
$$  
Simplify the numerator:  
$$
6x+8-2=6x+6.
$$  
Thus,  
$$
g(f(x))=\frac{6x+6}{3}.
$$  
Divide each term in the numerator by $$ 3 $$:  
$$
g(f(x))=2x+2.
$$
**a)** Calculate $$ g\left(\frac{1}{2}\right) $$: $$ g\left(\frac{1}{2}\right) = \frac{\frac{1}{2} - 2}{3} = -\frac{1}{2}. $$ **b)** Find $$ g(f(x)) $$: $$ g(f(x)) = \frac{6x + 8 - 2}{3} = 2x + 2. $$

The functions and are defined by
and
a) Calculate the value of .
b) Write an expression for in its simplest
form.

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The functions and are defined by and a) Calculate the value of . b) Write an expression for in its simplest form.