Question

Determine the graph of left-parenthesis f minus g right-parenthesis left-parenthesis x right-parenthesis when lower f left-parenthesis x right-parenthesis equals Start Fraction 1 over x End Fraction and lower g left parenthesis x right parenthesis equals Start Root x End Root.
A graph is shown on the coordinate plane. The x-axis goes from negative 20 to 20 and the y-axis goes from negative 20 to 20. The graph starts coming down along the y-axis and then curves down to the right crossing the x-axis and continues to curve to the right. There is an asymptote at x = 0.Image with alt text: A graph is shown on the coordinate plane. The x-axis goes from negative 20 to 20 and the y-axis goes from negative 20 to 20. The graph starts coming down along the y-axis and then curves down to the right crossing the x-axis and continues to curve to the right. There is an asymptote at x = 0.

A graph is shown on the coordinate plane. The x-axis goes from negative 15 to 35 and the y-axis goes from negative 5 to 15. The graph starts coming down along the y-axis. It begins to curve down to the right but then curves back up and to the right never crossing the x-axis. There is an asymptote at x = 0.
Image with alt text: A graph is shown on the coordinate plane. The x-axis goes from negative 15 to 35 and the y-axis goes from negative 5 to 15. The graph starts coming down along the y-axis. It begins to curve down to the right but then curves back up and to the right never crossing the x-axis. There is an asymptote at x = 0.

A graph is shown on the coordinate plane. The x-axis goes from negative 10 to 60 and the y-axis goes from negative 20 to 10. The graph starts coming up along the y-axis and then curves up to the right, crossing the x-axis and continuing to go up and to the right. There is an asymptote at x = 0.
Image with alt text: A graph is shown on the coordinate plane. The x-axis goes from negative 10 to 60 and the y-axis goes from negative 20 to 10. The graph starts coming up along the y-axis and then curves up to the right, crossing the x-axis and continuing to go up and to the right. There is an asymptote at x = 0.

A graph is shown on the coordinate plane. The x-axis goes from negative 10 to 60 and the y-axis goes from negative 20 to 10. The graph starts coming up along the y-axis. It begins to curve up to the right but then curves back down and to the right never crossing the x-axis. There is an asymptote at x = 0.

Answer 1

To find the graph of \((f - g)(x)\), we first need to define \(f(x)\) and \(g(x)\):

\[
f(x) = \frac{1}{x}
\]
\[
g(x) = \sqrt{x}
\]

Now, we determine \((f - g)(x)\):

\[
(f - g)(x) = f(x) - g(x) = \frac{1}{x} - \sqrt{x}
\]

To analyze this expression, let's consider the domains of \(f(x)\) and \(g(x)\):

- \(f(x)\) is defined for \(x \neq 0\) and can take any real value except when \(x = 0\).
- \(g(x)\) is defined for \(x \geq 0\).

Thus, the domain of \((f - g)(x)\) will be \(x > 0\) since \(g(x)\) is only defined for non-negative \(x\) and \(f(x)\) is not defined at \(x = 0\).

Next, we can consider the behavior of \((f - g)(x)\):

1. For very small positive values of \(x\):
- As \(x\) approaches \(0\) from the right, \(\frac{1}{x}\) approaches \(+\infty\) while \(\sqrt{x}\) approaches \(0\). Thus, \((f - g)(x)\) approaches \(+\infty\).

2. For values of \(x\) approaching infinity:
- \(\frac{1}{x}\) approaches \(0\) while \(\sqrt{x}\) approaches \(+\infty\). Therefore, \((f - g)(x)\) approaches \(-\infty\).

3. At \(x = 1\):
- \((f - g)(1) = 1 - 1 = 0\). This means the graph crosses the x-axis at \(x = 1\).

The graph will have an asymptote at \(x = 0\) and will start from \(+\infty\) and then cross the x-axis at \(x = 1\) and will continue to decrease, heading towards \(-\infty\) as \(x\) increases.

With this analysis, we can infer that the correct graph should start from the top left approaching the vertical asymptote at \(x = 0\), cross the x-axis at \(x = 1\), and continue downward towards \(y = -\infty\).

The description matches the first option you provided:
- "The graph starts coming down along the y-axis and then curves down to the right crossing the x-axis and continues to curve to the right. There is an asymptote at \(x = 0\)."

Therefore, the graph of \((f - g)(x)\) is the first graph described.