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