Question

consider the following function.
q(x)=\begin{cases}\frac{1}{2x}&\text{if }x < 2\\-\frac{4}{3}x&\text{if }x>2end{cases}
step 1 of 3: identify the general shape and direction of the graph of this function on the interval ((-infty,2)).
answer

Explanation:

Step1: Analyze the function for $x < 2$

The function is $q(x)=\frac{1}{2x}$. This is a rational - function of the form $y = \frac{k}{x}$ where $k=\frac{1}{2}$.

Step2: Determine the shape and direction

For a rational function $y=\frac{k}{x}$, when $k>0$, in the interval where it is defined, the graph is a hyperbola. For $q(x)=\frac{1}{2x}$ with $x\in(-\infty,2)$, as $x$ approaches $0$ from the left, $q(x)\to-\infty$, and as $x$ approaches $0$ from the right, $q(x)\to+\infty$. Also, as $x\to-\infty$, $q(x)\to0$ from the negative side. The function is decreasing on the interval $(-\infty,0)$ and $(0,2)$.

Answer:

The graph is a hyperbola and is decreasing on the intervals $(-\infty,0)$ and $(0,2)$.