I am trying to understand this question:
Use linear scales for both the $X$- and $Y$- axes. In which region, $f_2(x)$ is a good approximation to $f_1(x)$?
My equations are: $$f_1(x)=\sqrt{\frac{1+2x}{1+x}}\quad\text{and}\quad f_2(x)=1+0.5x-(0.625)x^2+(0.86667)x^3$$ graph of two functions Am I supposed to use: $$f(x_0) + f'(x_0)(x_1-x_0) = 0\Rightarrow x_1-x_0 = \frac{f(x_0)}{f'(x_0)} \Rightarrow x_1 = x_0-\frac{f(x_0)}{f'(x_0)}.$$ How do I find the best region?
I used Wolfram Alpha to make a graph of your two functions. Here is a version of that graph for $x$ values $-0.5 \leq x \leq 1$:
Note that $\sqrt{(1+2x)/(1+x)}$ is not even defined for $x < -0.5$, so there's not much point in looking any further to the left. But it is very important not to just automatically cut off the entire negative $x$-axis from a graph like this.
Since each function's curve is drawn in a different color, if you place a vertical straightedge against the graph and compare the $y$-values at which the two different-colored curves intersect the vertical line, you will have an idea of how good the approximation is at that point. Small or no gap is a good approximation, large gap is bad.
Both your graph and the graph above use linear scales for the $y$-axis, but not the same linear scale; the difference between the two curves will look smaller on the graph above merely because the scale is smaller. It is somewhat a matter of judgment to say what approximation is "good enough"; I doubt your instructor expects everyone to describe exactly the same region where the approximation is good. But you can see in either graph that part of the curves of the two functions seem to be drawn practically right on top of each other, which should be a clue.
If you estimate the minimum and maximum "good" $x$ values by looking at the graph, you can use a calculator to check the values of the function at those $x$ values to see exactly how small or large the difference between the functions is.