Chapter 3.7, Problem 38HP

a.

To determine

To determine whether the graph of $f(x)=\{\begin{array}{l}-3x+4\text{ if }x<0\\ x-2\text{ if }x\ge 0\end{array}$ includes the point $\left(0,4\right)$ .

Explanation of Solution

Given:

Function: $f(x)=\{\begin{array}{l}-3x+4\text{ if }x<0\\ x-2\text{ if }x\ge 0\end{array}$

Concept used:

A piecewise defined function is a function defined by at least two equations (“pieces”), each of which applies to a different part of the domain.

When graphing a piecewise function, focus on where the changes in the graph occur.

Calculation for graph:

Consider the function $f(x)=\{\begin{array}{l}-3x+4\text{ if }x<0\\ x-2\text{ if }x\ge 0\end{array},$

Here, from $x$ -values less than 0, the graph is a straight line and from $x$ -values from 0 to $\infty$ , the graph is a straight line.

Values of x	Values of f (x)
$-2$	$10$
$-1$	$7$
$0$	$-2$
$1$	$-1$
$2$	$0$

By taking different values of x , the graph can be plotted.

Graph:

The graph of $f(x)=\{\begin{array}{l}-3x+4\text{ if }x<0\\ x-2\text{ if }x\ge 0\end{array}$ :

  

Interpretation:

From the graph, it can be observed that the point $\left(0,4\right)$ is not included in the graph.

The error made by Dylan is that he has considered 0 in the interval $\left(-\infty ,0\right)$ which is not true according to the function defined.

The value of $f(0)=-2$ .