Suppose that
𝑦
=
6
𝑥
5
−
7
. Find the inverse, where
𝑥
=
𝑓
−
1
(
𝑦
)
.

Question

Suppose that 
𝑦
=
6
𝑥
5
−
7
. Find the inverse, where 
𝑥
=
𝑓
−
1
(
𝑦
)
.

BlackTom AI · Accepted Answer

Swap the roles of x and y, then solve for the new y. Starting with y = 6x^5 − 7, interchange x and y to get:
x = 6y^5 − 7.

Isolate the y^5 term by adding 7 to both sides:
x + 7 = 6y^5.

Divide both sides by 6:
y^5 = (x + 7)/6.

Take the fifth root (the 1/5 power) of both sides to solve for y:
y = ((x + 7)/6)^(1/5).

Because the problem labels the inverse as x = f^{-1}(y), replace y by x and x by y in the final expression (or simply rename variables as needed). Writing the inverse in the requested form gives:
x = ((y + 7)/6)^(1/5).

Why this is correct: composing f with this inverse returns the original input. If you plug x = ((y+7)/6)^(1/5) into f(t) = 6t^5 − 7, you get 6·(((y+7)/6)^(1/5))^5 − 7 = 6·((y+7)/6) − 7 = y + 7 − 7 = y, so f(f^{-1}(y)) = y. The original function is one-to-one (odd power and positive coefficient make it strictly increasing), so an inverse exists for all real y; taking the real fifth root is valid for every real input.

Common mistakes to avoid: failing to swap variables, forgetting to divide by 6 before taking the fifth root, or taking the reciprocal exponent incorrectly.

Conclusion: the inverse is x = ((y + 7)/6)^(1/5), which matches the provided correct answer.

Suppose that 𝑦 = 6 𝑥 5 − 7 . Find the inverse, where 𝑥 = 𝑓 − 1 ( 𝑦 ) .单项选择题

类似问题

Select the function that has a well-defined inverse.

A={a,b,c,d} X={1,2,3,4} Each choice defines a function whose domain is A and whose target is X. Select the function that has a well-defined inverse.

Which of the following does NOT have an inverse function?

考虑由下式给出的函数 𝑓 ( 𝑥 ) = 3 + 2 𝑥 + 3 𝑥 − 11 . 下列哪个是……的逆运算？ 𝑓 ？

Find the domain and range of the inverse of the given function. f(x) =

Question text Find the inverse function of the exponential function [math: f(x)=2e5x−2+1]\displaystyle f(x)={2\,e^{5\,x-2}+1}. Note: Type [math: ln(a)] to enter [math: loge(a)]log_{e}(a). [math: f−1(x)=]f^{-1}(x)=[input] Check Question 7

Question text Find the inverse function of the exponential function [math: f(x)=6e2x−4+1]\displaystyle f(x)={6\,e^{2\,x-4}+1}. Note: Type [math: ln(a)] to enter [math: loge(a)]log_{e}(a). [math: f−1(x)=]f^{-1}(x)=[input] Check Question 7

更多留学生实用工具

智能学习助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单