To find the function $g(x)$ in terms of $f(x)$.

For each point $$(a,b)$$ on the graph $$y = f(x)$$ the point $$(3a-1,\frac{b}{2})$$ is plotted forming the graph of another function $$y = g(x)$$. The problem asks us to find the function $$g(x)$$ in terms of $$f(x)$$.

Also, we need to describe the transformation.

Since $$a$$ goes to $$3a-1$$, the inverse of the function will be $$x \to \frac{x+1}{3}$$.

Hence the required function is $$g(x) = \frac{1}{2}f\left( \frac{x+1}{3}\right).$$

Note that $$g(3a-1) = \frac{1}{2}f\left( \frac{3a-1+1}{3}\right) = \frac{1}{2}f(a) = \frac{b}{2}.$$

Description of the transformation:

$$\bullet$$ Horizontal stretch by a factor of $$3$$.

$$\bullet$$ Vertical stretch by a factor of $$\frac{1}{2}$$.

$$\bullet$$ Horizontal translation by $$-1$$.

Is the solution correct?

Let $$X=3a-1\;\text{ and }\; g(X)=\frac b2$$
$$b=f(a)$$ or
$$2g(X)=f(\frac{X+1}{3})$$ So
$$g(X)=\frac 12f(\frac{X+1}{3})$$