QuestionDetermine the inverse of the following function: $$f(x)=3^{x-1}+4$$
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Studdy Solution
Verify the inverse function by checking if $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.
First, verify $f(f^{-1}(x)) = x$:
$$f(f^{-1}(x)) = f(\log_3(x - 4) + 1)$$ $$= 3^{(\log_3(x - 4) + 1) - 1} + 4$$ $$= 3^{\log_3(x - 4)} + 4$$ $$= (x - 4) + 4$$ $$= x$$
Now, verify $f^{-1}(f(x)) = x$:
$$f^{-1}(f(x)) = f^{-1}(3^{x-1} + 4)$$ $$= \log_3((3^{x-1} + 4) - 4) + 1$$ $$= \log_3(3^{x-1}) + 1$$ $$= (x-1) + 1$$ $$= x$$
Both verifications confirm the correctness of the inverse function.
The inverse function is:
$$f^{-1}(x) = \log_3(x - 4) + 1$$