Question

What is the inverse of:$f(x)=x^3+5$

Answer 1

Answer:

$\displaystyle \large{ {f}^{ - 1} (x) = \sqrt[3]{x - 5} }$

Step-by-step explanation:

To find an inverse, we swap x to f(x)/y and f(x)/y to x.

We are given:-

$\displaystyle \large{f(x) = {x}^{3} + 5}$

Swap:-

To make the equation look better and easier to simplify, we will be changing f(x) to y.

Thus:-

$\displaystyle \large{y= {x}^{3} + 5} \\ \displaystyle \large{x= {y}^{3} + 5}$

Now simplify to y-isolated.

Subtract both sides by 5.

$\displaystyle \large{x - 5= {y}^{3} + 5 - 5} \\ \displaystyle \large{x - 5= {y}^{3} }$

Cube root both sides.

$\displaystyle \large{ \sqrt[3]{x - 5} = \sqrt[3]{ {y}^{3} } } \\ \displaystyle \large{ \sqrt[3]{x - 5} = y }$

Convert y to f(x) and add exponent of -1 between f and (x).

$\displaystyle \large{ \sqrt[3]{x - 5} = {f}^{ - 1} (x) }$

To indicate that the function is an inverse.

And we're done!

Answer 2

Answer:

f⁻¹(x) = ∛x-5

Step-by-step explanation:

y = x³ + 5

x = y³ + 5  ... replace x with y

y³ = x-5

y = ∛x-5