Which function is graphed below
The Correct Answer and Explanation is:
The graph shown represents the function: f(x)=1xf(x) = \frac{1}{x}
Explanation
The graph provided is a classic representation of a reciprocal function, which generally takes the form: f(x)=1xnf(x) = \frac{1}{x^n}
In this specific case, the function is f(x)=1xf(x) = \frac{1}{x}, which is evident from several key characteristics shown in the graph.
1. Shape and Behavior
The curve is decreasing throughout the positive x-values and increasing negatively as x becomes more negative. This “L”-shaped behavior in the first and third quadrants is a hallmark of the reciprocal function. However, this particular graph only shows the first quadrant (positive x-values), which suggests the function shown is only for positive values of xx — meaning the domain is x>0x > 0.
2. Vertical and Horizontal Asymptotes
- The graph never touches the x-axis, but it approaches it as x→∞x \rightarrow \infty. This behavior indicates a horizontal asymptote at y=0y = 0.
- The graph rises steeply as it approaches x=0+x = 0^+, suggesting a vertical asymptote at x=0x = 0.
These asymptotic behaviors are consistent with f(x)=1xf(x) = \frac{1}{x}.
3. Key Values
If you test a few x-values:
- f(1)=11=1f(1) = \frac{1}{1} = 1
- f(2)=12=0.5f(2) = \frac{1}{2} = 0.5
- f(0.5)=10.5=2f(0.5) = \frac{1}{0.5} = 2
- f(0.25)=4f(0.25) = 4
- f(0.125)=8f(0.125) = 8
These y-values match what is seen on the graph, confirming the function plotted is decreasing and rapidly approaching infinity as x approaches zero from the right.
Conclusion
Based on the curve’s shape, asymptotes, and evaluated points, the function graphed is: f(x)=1x\boxed{f(x) = \frac{1}{x}}
