Question

55-58 The graph of a function f is shown. (The dashed lines indicate horizontal asymptotes). Find each of the following for the given function g.

a) The domain of g and \(g'\)

b) The critical numbers of g

c) The approximate value of \(g'\left( {\bf{6}} \right)\)

d) All vertical and horizontal asymptotes of g

56. \(g\left( x \right) = \left| {f\left( x \right)} \right|\)

Short answer

a) The domain of g is \(\left( { - \infty ,\infty } \right)\) , and domain of \(g'\) is \(\left( { - \infty ,3} \right) \cup \left( {3,7} \right) \cup \left( {7,\infty } \right)\).

b) The critical numbers of g are 3, 5, 7, and 9.

c) \( - 2\)

d) The curve of g has horizontal asymptote \(y = 2\) and \(y = 1\). There is no vertical asymptote.

Step-by-step solution

Find an answer for part (a)

The domain of g is\(\left( { - \infty ,\infty } \right)\).

The domain of\(g'\)is equal to the domain of\(f'\)except for values such that\(f\left( x \right) = 0\)and\(f'\left( x \right) = 0\).

As \(f'\left( 3 \right)\) does not exist and \(f\left( 7 \right) = 0\), and \(f'\left( 7 \right) \ne 0\). The domain of \(g'\) is \(\left( { - \infty ,3} \right) \cup \left( {3,7} \right) \cup \left( {7,\infty } \right)\).

Find an answer for part (b)

If\(g'\left( x \right) = 0\), then\(f'\left( x \right) = 0\).

From the figure, it can be observed that at\(x = 5\)and\(x = 9\), there are horizontal tangent lines.\(g'\left( x \right)\)does not exist at\(x = 3\)and\(x = 7\).

So, the critical numbers are 3, 5, 7, and 9.

Find an answer for part (c)

Since f is positive near \(x = 6\), \(g\left( x \right) = \left| {f\left( x \right)} \right|\), so, \(g'\left( 6 \right) = f'\left( 6 \right) \approx - 2\).

So, the value of \(g'\left( 6 \right)\) is \( - 2\).

Find an answer for part (d)

It can be observed from the graph that there is no vertical asymptote.

The horizontal asymptotes are:

\(\begin{array}{c}\mathop {\lim }\limits_{x \to - \infty } g\left( x \right) = \mathop {\lim }\limits_{x \to - \infty } \left| {g\left( x \right)} \right|\\ = \left| 2 \right|\\ = 2\end{array}\)

And,

\(\begin{array}{c}\mathop {\lim }\limits_{x \to \infty } g\left( x \right) = \mathop {\lim }\limits_{x \to \infty } \left| {f\left( x \right)} \right|\\ = \left| { - 1} \right|\\ = 1\end{array}\)

So, the horizontal asymptotes are \(y = 2\) and \(y = 1\).