Question

5-10 Sketch the graph of an example of a function f that satisfies all of the given conditions.

\(\mathop {{\bf{lim}}}\limits_{x \to - \infty } f\left( x \right) = - {\bf{1}}\),\(\mathop {{\bf{lim}}}\limits_{x \to {{\bf{0}}^ - }} f\left( x \right) = \infty \),\(\mathop {{\bf{lim}}}\limits_{x \to {{\bf{0}}^ + }} f\left( x \right) = - \infty \), \(\mathop {{\bf{lim}}}\limits_{x \to {{\bf{3}}^ - }} f\left( x \right) = {\bf{1}}\),\(f\left( {\bf{3}} \right) = {\bf{4}}\), \(\mathop {{\bf{lim}}}\limits_{x \to {{\bf{3}}^ + }} f\left( x \right) = {\bf{4}}\), \(\mathop {{\bf{lim}}}\limits_{x \to \infty } f\left( x \right) = {\bf{1}}\)

Short answer

The graph of an example of a function f is shown below:

Step-by-step solution

Step 1:Check the nature of the graph from the given values

It appears from the given values that there should be only twohorizontal asymptotes\(y = - 1\) and \(y = 1\).

As xis approaching 3 from the right, the value tends to 4, and when x is approaching 3 from the left, the value \(f\left( x \right)\)tends to 1.

So, the graph will be drawn in three sections.

Sketch the graph of \(f\left( x \right)\)

The graph of an example of a function f is shown below:

Thus, the graph is obtained.