Answer
• a) Range: (-∞, –1)
• b) Horizontal asymptote when x ≤ 2: y = –3. Vertical asymptote when x > 2: x= 2. Horizontal asymptote when x > 2: y = –1.
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• c) When x decreases, it approaches –3, and when x increases, it approaches –1.
Explanation
• Part A: Graph the piecewise function f (x) and determine the range.
Using a graphing tool we can get the following graph:
As the range is all the y-values included in the function, we can see that in this case, the range is:
[tex](-\infty,-1)[/tex]
• Part B: Determine the asymptotes of f (x). Show all necessary calculations.
We have to get the asymptotes from the part where x ≤ 2 and x > 2.
In x ≤ 2 we do not have a vertical asymptote as it is not a rational function, but we do have a horizontal asymptote, which is y = k when we have the function:
[tex]y=a(b)^{x-h}+k[/tex]
Thus, in our case y = –3.
In x > 2, we do have a vertical and horizontal asymptote. We can get the vertical asymptote by setting the denominator of the function to 0:
[tex]x^2-5x+6=0[/tex]
If we solve the expression by factoring we get two solutions:
[tex](x-2)(x-3)=0[/tex][tex]\begin{gathered} x_1-2=0 \\ x_1=2 \end{gathered}[/tex][tex]\begin{gathered} x_2-3=0 \\ x_2=3 \end{gathered}[/tex]
Based on our function we can see that the vertical asymptote is x = 2 when x > 2. Then, the horizontal asymptote can be calculated with the limit:
[tex]\lim_{x\to\infty}(\frac{-x^2+2x+3}{x^2-5x+6})=-1[/tex]
Thus, our horizontal asymptote is y = –1 when x > 2.
• Part C: Describe the end behavior of f (x).
The end behavior of a function is how the function acts when x increases or decreases. Based on our graph we can see that when x decreases, it approaches –3, and when x increases, it approaches –1.
