Explore a diverse range of topics and get answers from knowledgeable individuals on IDNLearn.com. Our community provides timely and precise responses to help you understand and solve any issue you face.

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]