Question

consider the following function.

w(x)=left{\begin{array}{ll}\frac{5}{2 x^{3}} & \text { if } x<2 \\ -\frac{7}{9} x & \text { if } x>2end{array}
ight.

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 for \(x < 2\) is \(w(x)=\frac{5}{2x^{3}}=\frac{5}{2}x^{- 3}\).

Step2: Recall the general form of power - functions

For a power - function \(y = ax^{n}\), when \(n<0\) (here \(n=-3\)) and \(a>0\) (here \(a = \frac{5}{2}\)), as \(x\) increases from \(-\infty\) to \(0\), \(y=\frac{5}{2x^{3}}\) approaches \(0\) from the negative side (\(x<0\), \(x^{3}<0\), so \(\frac{5}{2x^{3}}<0\)). As \(x\) increases from \(0\) to \(2\), \(y = \frac{5}{2x^{3}}\) is positive (\(x>0\), \(x^{3}>0\)) and decreases.

Answer:

The graph of \(y = w(x)\) on the interval \((-\infty,2)\) is a part of a power - function \(y=\frac{5}{2}x^{-3}\). It is in the form of a hyperbola - like curve. For \(x<0\), the function is negative and approaches \(0\) as \(x\to0^{-}\). For \(0 < x<2\), the function is positive and decreases as \(x\) increases.