Values $\theta$ such that $f+g$ and $g$ have the same domain, where $f(x)=\sqrt{\theta x^2-2(\theta^2-3)x-12\theta}$ and $g(x)=\ln(x^2-49)$

Question

Given are two functions

$$\begin{align} f(x)&=\sqrt{\theta x^2-2\left(\theta^2-3\right) x-12 \theta} \\ g(x)&=\ln \left(x^2-49\right) \end{align}$$

What is the range of $\theta$ such that functions $f+g$ and $g$ have same domain?

I tried applying $f(7)$ and $f(-7)$ $≤ 0$ as $D$ is greater than zero for given quadratic equation inside root. But I'm missing some more relations.

What are other those relations.

Answer 1

Domain of $g$ is $D_g=(-\infty,-7)\cup(7,\infty).$
This implies that the leading coefficient of the quadratic in $f(x)$ is positif. We can factor out $\theta.$ The domain $D_f$ is a set of solutions to $$x^2-2(\theta-\frac{3}{\theta})x-12>0,$$ which is $(-\infty,x_1)\cup(x_2,\infty)$ with $x_1,x_2\in\{-\frac{6}{\theta},2\theta\}.$

To fulfil $D_{f+g}=D_g,$ the system $$-7\leq -\frac{6}{\theta} \quad \text{and} \quad 2\theta\leq 7$$ has to be satisfied. This gives $$\theta\in\left[\frac{6}{7},\frac{7}{2}\right].$$