Science:Math Exam Resources/Courses/MATH110/April 2019/Question 06/Hint 1

The first bullet point tells us what the function looks like for x small enough or large enough: g(x) has a horizontal asymptote as $x\rightarrow -\infty$ and it should look go down to $-\infty$ as x increases. From the second bullet point, we can infer that g has a maximum at x=0. The graph of g away from x = -1 looks something like the graph of $e^{x}$ for $x<-1$ and the graph of $-x^{2}+1$ for $x>-1$.

A partial sketch is presented in figure 1. Note that there are many possible different sketches.

Figure 1: Partial sketch of a function g(x) satisfying the properties mentioned in question 6.