Answered

Which function has an asymptote at $x=5$ and an $x$-intercept of $(6,0)$?


A$f(x)=\log (x-5)$
B$f(x)=\log (x+5)$
C$f(x)=\log x-5$
D$f(x)=\log x+5$

Verified Answer :

  • Analyze the conditions for the vertical asymptote (\(x=5\)) and x-intercept (\((6,0)\)).
  • Test each function:
  • \(f(x) = \log(x - 5)\) has an asymptote at \(x = 5\) and intercept at \(\log(6-5) = \log(1) = 0\).
  • Other functions have asymptotes at \(x = -5\) or \(x = 0\).
  • Conclude that the correct function is \(\boxed{f(x) = \log(x - 5)}\).

Explanation

  1. Analysis of the Problem and Functions

We are given four candidate functions and asked to identify which one has a vertical asymptote at \(x = 5\) and an \(x\)-intercept at \((6, 0)\). The candidate functions are:

  1. \(f(x) = \log(x - 5)\)
  2. \(f(x) = \log(x + 5)\)
  3. \(f(x) = \log x - 5\)
  4. \(f(x) = \log x + 5\)

A vertical asymptote for a logarithmic function of the form \(f(x) = \log(g(x))\) occurs where the argument \(g(x)\) approaches \(0\) from the right. An \(x\)-intercept occurs at a value of \(x\) where the function evaluates to \(0\), i.e., \(f(x) = 0\).

  1. Evaluating Candidate Functions

We evaluate each function based on the required properties:

  • For \(f(x) = \log(x - 5)\):
  • Vertical Asymptote: The argument \(x - 5\) equals \(0\) when \(x = 5\). As \(x \to 5^+\), the value of \(\log(x-5)\) approaches \(-\infty\). This satisfies the requirement for a vertical asymptote at \(x = 5\).
  • \(x\)-intercept: Setting \(f(x) = 0\) gives \(\log(x - 5) = 0\), which means \(x - 5 = 10^0 = 1\), so \(x = 6\). This satisfies the requirement for an \(x\)-intercept at \((6, 0)\).
  • For \(f(x) = \log(x + 5)\):
  • Vertical Asymptote: The argument \(x + 5\) equals \(0\) when \(x = -5\). The asymptote is at \(x = -5\), not \(x = 5\).
  • For \(f(x) = \log x - 5\):
  • Vertical Asymptote: The argument \(x\) equals \(0\) when \(x = 0\). The asymptote is at \(x = 0\), not \(x = 5\).
  • For \(f(x) = \log x + 5\):
  • Vertical Asymptote: The argument \(x\) equals \(0\) when \(x = 0\). The asymptote is at \(x = 0\), not \(x = 5\).
  1. Conclusion

Based on our evaluation, the only function that satisfies both conditions—a vertical asymptote at \(x = 5\) and an \(x\)-intercept at \((6, 0)\)—is \(f(x) = \log(x - 5)\). The final answer is \(\boxed{f(x) = \log(x - 5)}\).

Examples

In engineering and data science, logarithmic functions are often used to model growth patterns that slow down over time or to compress data across a wide range of values. Understanding how vertical asymptotes shift is crucial when designing sensors or filters that must exclude certain low-intensity signals (the asymptote) while specifically calibrating the device so that a known baseline value (the x-intercept) corresponds to zero output. This ensures that the system provides accurate measurements within its defined operational range.

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