Which Of The Following Rational Functions Is Graphed Below

The graph below represents a rational function. To identify which specific function it depicts, we must analyze its key graphical features. Rational functions, expressed as the quotient of two polynomials, produce distinct patterns on a coordinate plane, primarily characterized by their asymptotes, intercepts, and behavior at infinity. By systematically examining these elements, we can pinpoint the exact function. Let's dissect the graph's critical characteristics step-by-step.

Key Features to Analyze:

1. Vertical Asymptotes: These are vertical lines (x = a) where the function approaches positive or negative infinity as x approaches a. They occur where the denominator is zero and the numerator is non-zero.
2. Horizontal Asymptotes: These are horizontal lines (y = b) that the function approaches as x approaches positive or negative infinity. They depend on the degrees of the numerator and denominator polynomials.
3. Holes (Removable Discontinuities): These are isolated points (x = a, y = b) where the function is undefined, but the limit exists. They occur where a factor in the numerator and denominator cancel out.
4. Intercepts: Points where the graph crosses the x-axis (x-intercepts) or y-axis (y-intercept).
5. End Behavior: How the graph behaves as x approaches ±∞.
6. Symmetry: Whether the graph is symmetric about the y-axis, x-axis, or origin.

Analyzing the Graph:

• Vertical Asymptotes: The graph shows vertical asymptotes at x = -3 and x = 2. This immediately tells us the denominator of the rational function must include factors (x + 3) and (x - 2). Therefore, the denominator is at least (x + 3)(x - 2).
• Horizontal Asymptote: The graph approaches a horizontal line at y = 1 as x approaches both +∞ and -∞. This indicates the degrees of the numerator and denominator polynomials are equal, and the ratio of the leading coefficients is 1 (since the asymptote is y = 1). If the leading coefficient of the numerator were 2, the asymptote would be y = 2. Here, it's clearly 1.
• Intercepts: The graph crosses the x-axis at x = 1. There is no y-intercept visible on the graph provided. The absence of a visible y-intercept doesn't necessarily mean there isn't one; it could be very close to the origin or obscured by the asymptote at x=2. However, the x-intercept at x=1 is a crucial point.
• Holes: There are no visible holes in the graph. The function is defined at every point shown.
• End Behavior: As described, the graph approaches y = 1 from above as x → -∞ and from below as x → +∞, consistent with the horizontal asymptote.
• Symmetry: The graph does not appear symmetric about the y-axis or the origin.

Putting it Together:

Based on the analysis:

1. The denominator must include factors (x + 3) and (x - 2), so it is at least (x + 3)(x - 2).
2. The degrees of the numerator and denominator are equal.
3. The ratio of the leading coefficients is 1, giving the horizontal asymptote y = 1.

The simplest function satisfying these conditions is a rational function where the numerator is a constant multiple of the denominator, specifically a constant equal to 1, since the leading coefficient ratio is 1. Therefore, the function is:

f(x) = (x + 3)(x - 2) / [(x + 3)(x - 2)]

However, this function simplifies to f(x) = 1 for all x except x = -3 and x = 2. This is a constant function with vertical asymptotes at x = -3 and x = 2. While mathematically correct, this constant function graph would be a horizontal line at y=1, with breaks at x=-3 and x=2. This doesn't match the graph showing an x-intercept at x=1.

Therefore, the numerator must be not a constant equal to the denominator. Since the degrees are equal and the leading coefficient ratio is 1, the numerator must be a polynomial of the same degree as the denominator, with leading coefficient 1. The simplest possibility is a linear numerator. Let's assume the numerator is a linear polynomial: (x - a).

The x-intercept is at x=1. This means f(1) = 0. Plugging x=1 into the function:

f(1) = [(1 - a)(1 + 3)] / [(1 + 3)(1 - 2)] = [(1 - a)(4)] / [(4)(-1)] = 4(1 - a) / (-4) = -(1 - a) = a - 1

Set this equal to zero for the x-intercept:

a - 1 = 0 => a = 1

So the numerator must be (x - 1). The denominator is (x + 3)(x - 2). This gives:

f(x) = (x - 1) / [(x + 3)(x - 2)]

Now, check the key features:

• Vertical Asymptotes: Denominator zero at x = -3 and x = 2. Numerator non-zero at these points. ✅ Matches.
• Horizontal Asymptote: Degrees equal (both degree 1), leading coefficient of numerator is 1, denominator is 1. Ratio is 1. Asymptote y=1. ✅ Matches.
• x-Intercept: Numerator zero at x=1, denominator non-zero. Graph crosses x-axis at (1,0). ✅ Matches (visible at x=1).
• y-Intercept: f(0) = (0 - 1) / [(0 + 3)(0 - 2)]