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Determine the inverse of the equation [tex]y = 3x^3 - 5[/tex].

A. [tex]y = \sqrt[3]{\frac{x+5}{3}}[/tex]
B. [tex]y = \sqrt[3]{\frac{x-5}{3}}[/tex]
C. [tex]y = 3 \sqrt[3]{x-5}[/tex]
D. [tex]y = \sqrt[3]{\frac{x}{3}} + 5[/tex]


Sagot :

To find the inverse of the function [tex]\( y = 3x^3 - 5 \)[/tex], we need to switch the roles of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] and then solve for [tex]\( y \)[/tex]. Here are the steps:

1. Replace [tex]\( y \)[/tex] with [tex]\( x \)[/tex] and [tex]\( x \)[/tex] with [tex]\( y \)[/tex]:

[tex]\[ x = 3y^3 - 5 \][/tex]

2. Solve for [tex]\( y \)[/tex]:

First, isolate the term involving [tex]\( y \)[/tex]:
[tex]\[ x + 5 = 3y^3 \][/tex]

Next, divide both sides by 3 to completely isolate [tex]\( y^3 \)[/tex]:
[tex]\[ \frac{x + 5}{3} = y^3 \][/tex]

Now, take the cube root of both sides to solve for [tex]\( y \)[/tex]:
[tex]\[ y = \sqrt[3]{\frac{x + 5}{3}} \][/tex]

So the inverse function is [tex]\( y = \sqrt[3]{\frac{x + 5}{3}} \)[/tex].

Now, let's verify which option matches this inverse function:

(A) [tex]\( y = \sqrt[3]{\frac{x + 5}{3}} \)[/tex]

(B) [tex]\( y = \sqrt[3]{\frac{x-5}{3}} \)[/tex]

(C) [tex]\( y = 3 \sqrt[3]{x-5} \)[/tex]

(D) [tex]\( y = \sqrt[3]{\frac{x}{3}} + 5 \)[/tex]

By comparing these options to our derived inverse function [tex]\( y = \sqrt[3]{\frac{x + 5}{3}} \)[/tex], it’s clear that option (A) is correct.

Therefore, the correct answer is:

[tex]\[ \boxed{A} \][/tex]
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