Question text Find the inverse function of the exponential function [math: f(x)=6e2x−4+1]\displaystyle f(x)={6\,e^{2\,x-4}+1}. Note: Type [math: ln(a)] to enter [math: loge(a)]log_{e}(a). [math: f−1(x)=]f^{-1}(x)=[input] Check Question 7

Question

BlackTom AI · Accepted Answer

We start by identifying the given function and setting up the inverse.
The function is f(x) = 6 e^{2x - 4} + 1. To find the inverse, we swap x and y and solve for y:

Let y = f(x) = 6 e^{2x - 4} + 1. Then y - 1 = 6 e^{2x - 4}.
Divide both sides by 6: (y - 1)/6 = e^{2x - 4}.
Take the natural logarithm of both sides: ln((y - 1)/6) = 2x - 4.
Add 4 to both sides: ln((y - 1)/6) + 4 = 2x.
Divide by 2: x = (1/2) ln((y - 1)/6) + 2.
Thus the inverse function is f^{-1}(y) = (1/2) ln((y - 1)/6) + 2.
Replacing y with x to express the inverse function in standard form: f^{-1}(x) = (1/2) ln((x - 1)/6) + 2.

Key checks and notes:
- The natural log requires its argument to be positive, so (x - 1)/6 > 0, which implies x > 1 for the domain of the inverse.
- We can also rewrite f^{-1}(x) = (1/2) [ln(x - 1) - ln 6] + 2 if desired, but the compact form (1/2) ln((x - 1)/6) + 2 is standard.
- The given answer matches this result exactly: (1/2) ln((x - 1)/6) + 2, confirming correctness.

Question text Find the inverse function of the exponential function [math: f(x)=6e2x−4+1]\displaystyle f(x)={6\,e^{2\,x-4}+1}. Note: Type [math: ln(a)] to enter [math: loge(a)]log_{e}(a). [math: f−1(x)=]f^{-1}(x)=[input] Check Question 7多项填空题

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