To find the inverse of the function [tex]\( f(x) = x^{\frac{3}{7}} - 10 \)[/tex], follow these steps:
1. Set [tex]\( y = f(x) \)[/tex]:
[tex]\[ y = x^{\frac{3}{7}} - 10 \][/tex]
2. Solve for [tex]\( x \)[/tex]:
- Start by isolating the term with [tex]\( x \)[/tex]. Add 10 to both sides:
[tex]\[ y + 10 = x^{\frac{3}{7}} \][/tex]
- To solve for [tex]\( x \)[/tex], raise both sides of the equation to the power of [tex]\( \frac{7}{3} \)[/tex] (since [tex]\((a^{\frac{3}{7}})^{\frac{7}{3}} = a\)[/tex]):
[tex]\[ (y + 10)^{\frac{7}{3}} = x \][/tex]
3. Write the inverse function:
- Replace [tex]\( y \)[/tex] with [tex]\( x \)[/tex] to express the inverse function:
[tex]\[ f^{-1}(x) = (x + 10)^{\frac{7}{3}} \][/tex]
Hence, the inverse of the function [tex]\( f(x) = x^{\frac{3}{7}} - 10 \)[/tex] is:
[tex]\[
f^{-1}(x) = (x + 10)^{\frac{7}{3}}
\][/tex]
