Exercise 1 Page 92

Let's start by thinking about how the calculator creates a graph. To do so, it must decide whether to color a pixel or leave it blank. The colored pixels together form the graph of the function.

It is important to note that each pixel represents a segment or set of values, not just a single value.

In Example $1,$ the calculator is graphing the functions shown below. These functions are undefined for $x>2.5$ and for $x<\text{-}2.5,$ since $2.5^2=6.25.$ For the first picture, where we have a gap, the $x$ values go from $\text{-}6$ to $6.$ Since we know that the calculator has $95$ pixels in its width, we can calculate the length in units that each pixel represents. Using the screen's left side as reference, we start from $\text{-}6.$ Then, we need to move $3.5$ units to get to $\text{-}2.5.$ All the pixels corresponding to the $x\text{-values}$ between these segment will be blank pixels, since they are outside the domain of both functions. We can find how many pixels will be used for covering this length by dividing the $3.5$ units distance by $0.1263,$ which is the length each pixel represent. So this is what is happening. The first $27$ pixels, starting from the left of the screen, are blank. Then, about $71\%$ of the $28^{\text{th}}$ pixel, represents $x\text{-values}$ outside the domain of both functions, and just $29\%$ of it would be valid values. Hence, the calculator decides to leave it blank.

These blank pixels create the gap we see in the graph.

On the other hand, in the picture where the gap problem is fixed, the $x\text{-values}$ go from $\text{-}4.7$ to $4.7.$ Again, this will be represented using the $95$ pixels on its width. Let's calculate the length in units that each pixel represents. Once more, using the screen's left size as reference, we start from $\text{-}4.7$ this time. We need to move $2.2$ units to get to $\text{-}2.5.$ Again, all the pixels corresponding to the $x\text{-values}$ between these segment will be blank pixels, since they are outside the domain of both functions. Let's see how many pixels these will be. Recall that each pixel represent approximately $0.0989$ units now. This time the first $24$ pixels, starting from the left of the screen, are blank. Then, about $24\%$ of the $25^{\text{th}}$ pixel, represents $x$ values outside the domain of the functions, while $76\%$ are valid values. Hence, the calculator decides to color the pixel.

This time, there is no gap because the pixels connecting the graphs have been colored.