Graphical Features. Analyze the function below

Question

Graphical Features. Analyze the function below and produce a detailed graph showing all the features identified in 4.5, specifically the domain, intercepts, asymptotes, extrema, and inflection points. Use exact values whenever possible.

G(x) = (x-2)*(e^(-1/x)) 

Answer 1

1. Domain: -∞ < 0 < ∞ ALSO CAN BE WRITTEN AS (-∞, 0) U (0, ∞)
2. The only restriction in the function where it is not defined is in the exponent which is -1/x.
3. Because we cannot divide by 0, x ≠ 0
4. This gives us our domain
5. Intercepts: (2,0) and no Y-intercept
6. X-intercept occurs when you set the function equal to 0 and solve for x.
7. Since e^(-1/x) is always greater than 0, x-2 is the only part that can equal 0, so x=2.
8. Y-intercept occurs when you set x equal to 0 and solve.
9. Since the exponent has 0 as the denominator, it is undefined and no Y-intercept.
10. Asymptotes: VA: x=0, HA: none, SA: y = x-3
11. Vertical asumptotes (VA) occur when you set the domain restrictions into limits.
12. The lim of G(x) as x approaches 0 from the left is -∞.
13. The lim of G(x) as x approaches 0 from the right is 0.
14. Because G(x) approaches -∞ from the left, there is a VA at x=0.
15. Horizontal asymptotes (HA) occur when the limit of G(x) as x approaches ±∞.
16. When x approaches ±∞, e^(-1/x) approaches 1.
17. When x approaches ±∞, (x-2) goes to ±∞, so there are none,
18. Slant aymptotes (SA) occur when the degree of the numerator is 1 more than the degree of the denominator.
19. They are written in the form y = mx + b.
20. First the slope, by rearranging the equation algebraically (ignoring b because we are taking a limit as G(x) divided by x goes to ∞), we get the slope as 1.
21. Now, b is calculated by seeing the limit of G(x) - mx as x goes to ∞. You get b is equal to -3.
22. Extrema: Local max: (-2, -4√e) Local min: (1, -1/e)
23. To find extrema, take the first derivative and see when it equals 0 or DNE.
24. Use the power, chain, product, and quotient rule to get the first derivative which is e^(-1/x) times (x+2) times (x-1) and that is all over x^2.
25. Now, find the critical points by setting that equal to 0 and DNE and you get x = -2, 1 and 0.
26. Now make a first derivative sign chart and see when G'(x) is positive and negative.
27. In the intervals (-∞,-2), (-2,0), (0,1), and (1,∞), the functions are positive, negative, negative, and positive, respectfully.
28. Local maximums happen when the derivative changes from positive to negative which occurs ar x = -2, plug that into the original function to get the y value.
29. Local minimums happen when the derivative changes from negative to positive which occurs ar x = 1, plug that into the original function to get the y value.