Question:
The function $f : R → R, f (x)$ is a polynomial function of degree 4. Part of the graph of $f$ is shown below. The graph of $f$ touches the x-axis at the origin.
part a) wants me to find the rule of $f$, which I did:
$f(x) = -4x^2\left(x^2-1\right)$
It then asks for
Let $g$ be a function with the same rule as $f$.
Let $h : D → R, h (x) =$ $ln\left(g\left(x\right)\right)-ln\left(x^3+x^2\right)$, where $D $is the maximal domain of $h$.
b) State $D$
So first what does it mean by "maximal domain?" Does it mean that I have to find the domain in which the Maximum occurs?
c) State the range of $h$
So I guess I have to simplify $ln(-4x^2\left(x^2-1\right))-ln\left(x^3+x^2\right)$, and so I did.
$h(x) = ln\left(-4\left(x-1\right)\right)$
I keep trying by have no success. What is the general approach when solving for domain /range? What are the things that I should keep in mind?
The equation $\quad y = -4x^2(x^2-1)\implies x\in\mathbb{R}$
Solving for $x$, we get $$x = \pm\frac{\sqrt{1 \pm \sqrt{1 - y}}}{\sqrt{2}}\implies -\infty \le y\le1$$ Any $y>1$ results in complex numbers. This form also suggests that $-\infty \le x \le \infty$ because the square root of [a number approaching] infinity is $\pm$ infinity.