Answered

Which exponential function does not have an $x$-intercept?


A$f(x)=5^{x-5}-1$
B$f(x)=5^{x-5}-5$
C$f(x)=-5^{x-5}+5$
D$f(x)=-5^{x-5}-1$

Verified Answer :

  • An x-intercept exists when \(f(x) = 0\).
  • Testing \(f_1(x) = 5^{x-5} - 1\): \(5^{x-5} = 1 \implies x = 5\).
  • Testing \(f_2(x) = 5^{x-5} - 5\): \(5^{x-5} = 5 \implies x = 6\).
  • Testing \(f_3(x) = -5^{x-5} + 5\): \(5^{x-5} = 5 \implies x = 6\).
  • Testing \(f_4(x) = -5^{x-5} - 1\): \(5^{x-5} = -1\), which has no solution since \(5^{x-5} > 0\).
  • The function with no x-intercept is \(\boxed{f(x) = -5^{x-5} - 1}\).

Explanation

  1. Analysis of x-intercepts for Exponential Functions

To determine which of the provided exponential functions does not have an x-intercept, we need to analyze the condition for an x-intercept. An x-intercept occurs for a function \(f(x)\) when there exists a real number \(x\) such that \(f(x) = 0\). We are given four functions:
\(f_1(x) = 5^{x-5} - 1\)
\(f_2(x) = 5^{x-5} - 5\)
\(f_3(x) = -5^{x-5} + 5\)
\(f_4(x) = -5^{x-5} - 1\)
An exponential term of the form \(b^{x-h}\) (where \(b > 0\)) is always strictly positive, meaning \(b^{x-h} > 0\) for all real \(x\). We will examine each function by setting it to zero and checking for solutions.

  1. Testing Each Function

We test each function individually:

  1. For \(f_1(x) = 5^{x-5} - 1 = 0\):

\(5^{x-5} = 1\)
Since \(1 = 5^0\), we have \(5^{x-5} = 5^0\), which implies \(x-5 = 0\), so \(x = 5\). This function has an x-intercept.

  1. For \(f_2(x) = 5^{x-5} - 5 = 0\):

\(5^{x-5} = 5\)
Since \(5 = 5^1\), we have \(5^{x-5} = 5^1\), which implies \(x-5 = 1\), so \(x = 6\). This function has an x-intercept.

  1. For \(f_3(x) = -5^{x-5} + 5 = 0\):

\(-5^{x-5} = -5\)
\(5^{x-5} = 5\)
Since \(5 = 5^1\), we have \(5^{x-5} = 5^1\), which implies \(x-5 = 1\), so \(x = 6\). This function has an x-intercept.

  1. For \(f_4(x) = -5^{x-5} - 1 = 0\):

\(-5^{x-5} = 1\)
\(5^{x-5} = -1\)
Because the value of \(5^{x-5}\) must be greater than \(0\) for any real number \(x\), there is no value of \(x\) that satisfies the equation \(5^{x-5} = -1\). Therefore, this function does not have an x-intercept.

  1. Conclusion

Based on our analysis, we have identified that \(f_1(x)\), \(f_2(x)\), and \(f_3(x)\) each have an x-intercept, while the equation \(f_4(x) = 0\) yields no real solution for \(x\). Consequently, the function \(f(x) = -5^{x-5} - 1\) does not have an x-intercept. The final answer is \(\boxed{f(x) = -5^{x-5} - 1}\).

Examples

Understanding x-intercepts is crucial in real-world modeling. For example, if you are modeling the cooling of a cup of coffee using an exponential decay function, the x-intercept would represent the time at which the coffee reaches room temperature. If an equation has no x-intercept, it might suggest that the system being modeled never reaches a specific target value or state (e.g., the coffee might approach room temperature asymptotically but never actually reach it mathematically). Recognizing these mathematical constraints helps researchers interpret the behavior of real-world phenomena accurately.

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