Relationship Between a Linear Function and its Reciprocal
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- In a reciprocal function of the form \(y = \frac{1}{f(x)}\), vertical asymptotes occur when \(f(x) = 0\). So, a vertical asymptote will pass through the \(x\)-intercept of the graph of \(y = f(x)\).
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A reciprocal function of the form \(y = \frac{1}{f(x)}\) cannot be equal to zero. The line \(y = 0\) is always a horizontal asymptote of functions of this form.
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The graphs of \(y = f(x)\) and its corresponding reciprocal function \(y = \frac{1}{f(x)}\) intersect when the \(y\)-values are \(\pm 1\). These are the invariant points.