Verified Answer :

  • Set the two functions equal: \(2^x + 1 = -x + 7\), which simplifies to \(2^x + x = 6\).
  • Define \(h(x) = 2^x + x\) and observe it is strictly increasing.
  • Test integer values to find \(h(2) = 2^2 + 2 = 6\).
  • Verify the solution: \(f(2) = 2^2 + 1 = 5\) and \(g(2) = -2 + 7 = 5\).
  • The solution to the equation is \(x = \boxed{2}\).

Explanation

  1. Problem Analysis and Equation Setup

We are given two functions: \(f(x) = 2^x + 1\) and \(g(x) = -x + 7\). We need to find the value of \(x\) where these two functions intersect, which means solving the equation \(f(x) = g(x)\). Substituting the given expressions, we have:
\[2^x + 1 = -x + 7\]
To simplify, we isolate the terms involving \(x\) on one side:
\[2^x + x = 7 - 1\]
\[2^x + x = 6\]

  1. Solving for x

Let \(h(x) = 2^x + x\). We want to find the value of \(x\) such that \(h(x) = 6\).
Note that \(h(x)\) is the sum of an exponential function \(2^x\) and a linear function \(x\). Since both \(2^x\) and \(x\) are strictly increasing, \(h(x)\) must also be strictly increasing. This guarantees that there is at most one unique solution to the equation \(h(x) = 6\).
We can test integer values for \(x\):

  • For \(x = 1\): \(h(1) = 2^1 + 1 = 3\). Since \(3 < 6\), we need a larger \(x\).
  • For \(x = 2\): \(h(2) = 2^2 + 2 = 4 + 2 = 6\).

Since \(h(2) = 6\), the solution is \(x = 2\).

  1. Verification and Conclusion

To confirm our result, we check the intersection point by evaluating both functions at \(x = 2\):

  • \(f(2) = 2^2 + 1 = 4 + 1 = 5\).
  • \(g(2) = -2 + 7 = 5\).

Since \(f(2) = g(2) = 5\), the point of intersection is \((2, 5)\). Thus, the solution to the equation \(f(x) = g(x)\) is \(\boxed{2}\).

Examples

Determining the point of intersection between an exponential function and a linear function is essential in various fields, such as finance or population studies. For instance, if a company's revenue grows exponentially while its costs decrease linearly, solving \(f(x) = g(x)\) allows the company to calculate the exact time \(x\) when the revenue and costs reach the break-even point. Finding such equilibrium points helps in making informed decisions about scaling and growth strategies.

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