To find the inverse of the function [tex]\( f(x) = x^{\frac{1}{7}} - 10 \)[/tex], we will follow these steps:
1. Replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[
y = x^{\frac{1}{7}} - 10
\][/tex]
2. Solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
- First, isolate the term involving [tex]\( x \)[/tex]:
[tex]\[
y + 10 = x^{\frac{1}{7}}
\][/tex]
- Next, raise both sides to the power of 7 to solve for [tex]\( x \)[/tex]:
[tex]\[
(y + 10)^7 = x
\][/tex]
3. Express the inverse function [tex]\( f^{-1}(x) \)[/tex]:
- Replace [tex]\( y \)[/tex] with [tex]\( x \)[/tex] to find the inverse function:
[tex]\[
f^{-1}(x) = (x + 10)^7
\][/tex]
Hence, the inverse function [tex]\( f^{-1}(x) \)[/tex] is:
[tex]\[
f^{-1}(x) = (x + 10)^7
\][/tex]
So, [tex]\( f^{-1}(x) = 10000000.0 \cdot (0.1 \cdot x + 1)^7 \)[/tex], which simplifies our answer as:
[tex]\[
f^{-1}(x) = (x + 10)^7
\][/tex]
You can see that both formulations represent the same relationship, confirming the correctness of the inverse function.
