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?
