Question

Find the inverse function of the function f(x) = 2x.

NEED ASAP

Answer 1

as you already know, to get the inverse of any expression we start off by doing a quick switcheroo on the variables and then solving for "y", let's do so.

`\stackrel{f(x)}{y}~~ = ~~2\sqrt[3]{x}\hspace{5em}\stackrel{\textit{quick switcheroo}}{x~~ = ~~2\sqrt[3]{y}} \\\\\\ \cfrac{x}{2}=\sqrt[3]{y}\implies \left( \cfrac{x}{2} \right)^3=y\implies \cfrac{x^3}{2^3}=y\implies \cfrac{x^3}{8}=y~~ = ~~f^(-1)(x)`

Answer 2

f⁻¹(x) = x³/8

Explanation:

To find the inverse of the function f(x) = 2∛x, we will follow a systematic approach. This involves expressing the function in terms of 'y', switching the roles of 'x' and 'y', and then solving for 'y' to obtain the inverse function.

`\hrulefill`

Step 1: Express the Function in Terms of 'y'.

⇒ f(x) = 2∛x

⇒ y = 2∛x

Step 2: Interchange 'x' and 'y'.

⇒ y = 2∛x

⇒ x = 2∛y

Step 3: Solve for 'y'

⇒ x = 2∛y

⇒ x/2 = ∛y (Divide by 2)

⇒ (x/2)³ = (∛y)³ (Cube both sides)

⇒ y = (x/2)³ (Simplifying...)

⇒ y = x³/2³ (Simplifying...)

⇒ y = x³/8 (Simplified)

∴ f⁻¹(x) = x³/8