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The one-to-one function [tex]$f$[/tex] is defined below:

[tex]\[ f(x)=\sqrt[3]{7x+6} \][/tex]

Find [tex]$f^{-1}(x)$[/tex], where [tex]$f^{-1}$[/tex] is the inverse of [tex][tex]$f$[/tex][/tex].

[tex]\[ f^{-1}(x) = \boxed{\phantom{answer}} \][/tex]

Sagot :

GinnyAnsweridn
To find the inverse function [tex]\( f^{-1}(x) \)[/tex] for [tex]\( f(x) = \sqrt[3]{7x + 6} \)[/tex], follow these steps:

1. Express [tex]\( y = f(x) \)[/tex]:
[tex]\[ y = \sqrt[3]{7x + 6} \][/tex]

2. Solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
[tex]\[ y = (7x + 6)^{1/3} \][/tex]

3. Cube both sides of the equation to eliminate the cube root:
[tex]\[ y^3 = 7x + 6 \][/tex]

4. Isolate [tex]\( x \)[/tex] on one side of the equation:
[tex]\[ y^3 - 6 = 7x \][/tex]

5. Divide both sides by 7 to solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{y^3 - 6}{7} \][/tex]

6. Express the inverse function, [tex]\( f^{-1}(x) \)[/tex], using the simplified form:
[tex]\[ f^{-1}(x) = \frac{x^3 - 6}{7} \][/tex]

7. Simplify the fraction:
[tex]\[ f^{-1}(x) = \frac{1}{7} x^3 - \frac{6}{7} \][/tex]

So, the inverse function is:
[tex]\[ f^{-1}(x) = 0.142857142857143x^3 - 0.857142857142857 \][/tex]
This is the detailed step-by-step process for finding the inverse function [tex]\( f^{-1}(x) \)[/tex].
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