Exponential–log function, area and antiderivative

Find $D$.

Show that $f$ can be expressed in the form $f : D \to \R, f(x) = 4x^3$.

Find the area bounded by the graph of $y = f(x)$, the $x$-axis and the lines $x = 1$ and $x = 2$.

The function $g$ is the anti-derivative of $f$ that passes through the point $(1, -5)$.

Sketch the function $g$ on the axes below, over the domain $D$.

Label the coordinates of the endpoint, the $x$-intercept and the points on the graph where $x = 1$ and $x = 2$.

Let $a > 0.5$

The area bounded by the graph of $y = f(x)$, the $x$-axis and the lines $x = 0.5$ and $x = a$ can be expressed in the form $g(a) + b$.

Find the value of $b$.

Consider the function $h : (1.5, \infty) \to \R, h(x) = f(2x - 3) - f(x)$.

Find a quadratic expression for $h'(x)$.