‘Which of the following rational functions is graphed below? (x+1) 0 ^ Fx) = (x-1(+2) 2x 0 B Fx) = xx-2) (x1) 0 c Fx) = x(x-2) (x+1) 0 D. Fx) (x+ 2(* – 2) PREVIOUS’
The Correct Answer and Explanation is:
To determine which rational function is represented by the graph, we will analyze the key features of the graph: vertical asymptotes, horizontal asymptotes, and intercepts.
1. Identify the Vertical Asymptotes:
Vertical asymptotes occur where the function is undefined, which corresponds to the values of x that make the denominator of the rational function equal to zero.
Looking at the graph, we can see two vertical dashed lines, which represent the vertical asymptotes.
- One asymptote is at x = -2.
- The other asymptote is at x = 2.
These asymptotes tell us that the denominator of the function must have the factors (x – (-2)) and (x – 2), which are (x + 2) and (x – 2).
2. Identify the x-intercept(s):
An x-intercept is a point where the graph crosses the x-axis. At this point, the value of the function F(x) is 0. This happens when the numerator of the rational function is zero (and the denominator is not).
The graph crosses the x-axis at x = -1.
This means the numerator of the function must have a factor of (x – (-1)), which is (x + 1).
3. Identify the Horizontal Asymptote:
A horizontal asymptote describes the behavior of the function as x approaches positive or negative infinity.
The graph shows that as x gets very large in both the positive and negative directions, the curve approaches the x-axis.
Therefore, the horizontal asymptote is the line y = 0.
For a rational function, the horizontal asymptote is y = 0 when the degree of the numerator is less than the degree of the denominator.
4. Assemble the Function and Verify:
Based on our analysis, the function should have the form:
F(x) = (x + 1) / [(x + 2)(x – 2)]
Let’s check if this fits all our observations:
- Vertical Asymptotes: The denominator (x+2)(x-2) is zero at x = -2 and x = 2. This matches.
- x-intercept: The numerator (x+1) is zero at x = -1. This matches.
- Horizontal Asymptote: The degree of the numerator is 1, and the degree of the denominator is 2. Since 1 < 2, the horizontal asymptote is y = 0. This matches.
- y-intercept: We can also check the y-intercept. Let x = 0:
F(0) = (0 + 1) / [(0 + 2)(0 – 2)] = 1 / (2 * -2) = -1/4.
The graph appears to cross the y-axis at (0, -1/4), which further confirms our function is correct.
Conclusion:
The function F(x) = (x + 1) / [(x + 2)(x – 2)] perfectly matches the graph. Comparing this with the given choices, we find that it corresponds to option D.
The correct answer is D. F(x) = (x + 1) / [(x + 2)(x – 2)].
