Asked by Khaled Bushehri on Jun 09, 2024
Verified
Find the inverse function of f(t) =t7−5f ( t ) = t ^ { 7 } - 5f(t) =t7−5 .
A) f−1(t) =t−57f ^ { - 1 } ( t ) = \sqrt [ 7 ] { t - 5 }f−1(t) =7t−5
B) f−1(t) =t7−5f ^ { - 1 } ( t ) = \sqrt [ 7 ] { t } - 5f−1(t) =7t−5
C) f−1(t) =t+57f ^ { - 1 } ( t ) = \sqrt [ 7 ] { t + 5 }f−1(t) =7t+5
D) f−1(t) =t7+5f ^ { - 1 } ( t ) = \sqrt [ 7 ] { t } + 5f−1(t) =7t+5
E) The function does not have an inverse.
Inverse Function
An inverse function reverses the operation of the original function, mapping the output back to its input.
Function
An alliance between a compilation of inputs and a list of legitimate outputs, where it's mandated that each input is attributed to only one output.
- Determine and explain the reciprocal of provided functions.
Verified Answer
BM
Bettina MateoJun 10, 2024
Final Answer :
C
Explanation :
To find the inverse function of $f(t)$, we need to solve the equation $f(x)=t$ for $x$. So we have
f(x)=tx7−5=tx7=t+5x=t+57.\begin{align*}f(x)&=t\\x^7-5&=t\\x^7&=t+5\\x&=\sqrt[7]{t+5}.\end{align*}f(x)x7−5x7x=t=t=t+5=7t+5.
Thus, the inverse function is $f^{-1}(t)=\sqrt[7]{t+5}$. Option (C) matches our answer, so it is the correct choice.
f(x)=tx7−5=tx7=t+5x=t+57.\begin{align*}f(x)&=t\\x^7-5&=t\\x^7&=t+5\\x&=\sqrt[7]{t+5}.\end{align*}f(x)x7−5x7x=t=t=t+5=7t+5.
Thus, the inverse function is $f^{-1}(t)=\sqrt[7]{t+5}$. Option (C) matches our answer, so it is the correct choice.
Learning Objectives
- Determine and explain the reciprocal of provided functions.