Asked by demeris moore on Apr 26, 2024
Verified
Which statement(s) verify that f(x) =x+47f ( x ) = \sqrt [ 7 ] { x + 4 }f(x) =7x+4 and g(x) =x2−4g ( x ) = x ^ { 2 } - 4g(x) =x2−4 are inverse?
A) f(g(x) ) =x7−4+47=x;g(f(x) ) =(x7) 7+4−4=xf ( g ( x ) ) = \sqrt [ 7 ] { x ^ { 7 } - 4 + 4 } = x ; g ( f ( x ) ) = ( \sqrt [ 7 ] { x } ) ^ { 7 } + 4 - 4 = xf(g(x) ) =7x7−4+4=x;g(f(x) ) =(7x) 7+4−4=x
B) f(x) ⋅g(x) =x+47⋅(x7−4) =xf ( x ) \cdot g ( x ) = \sqrt [ 7 ] { x + 4 } \cdot \left( x ^ { 7 } - 4 \right) = xf(x) ⋅g(x) =7x+4⋅(x7−4) =x
C) f(g(x) ) =x77−4+4=x;g(f(x) ) =(x+47) 7−4=xf ( g ( x ) ) = \sqrt [ 7 ] { x ^ { 7 } } - 4 + 4 = x ; g ( f ( x ) ) = ( \sqrt [ 7 ] { x + 4 } ) ^ { 7 } - 4 = xf(g(x) ) =7x7−4+4=x;g(f(x) ) =(7x+4) 7−4=x
D) f(g(x) ) =x7−4+47=x;g(f(x) ) =(x+47) 7−4=xf ( g ( x ) ) = \sqrt [ 7 ] { x ^ { 7 } - 4 + 4 } = x ; g ( f ( x ) ) = ( \sqrt [ 7 ] { x + 4 } ) ^ { 7 } - 4 = xf(g(x) ) =7x7−4+4=x;g(f(x) ) =(7x+4) 7−4=x
E) f(x) g(x) =x+47x7−b=−1\frac { f ( x ) } { g ( x ) } = \frac { \sqrt [ 7 ] { x + 4 } } { x ^ { 7 } - b } = - 1g(x) f(x) =x7−b7x+4=−1
Inverse Function
A mathematical function that reverses the effect of another function, such that if the function f applied to an input x gives a result of y, then applying its inverse function to y gives the result x.
Function
An interrelation between a variety of inputs and a catalog of admissible outputs, with the requirement that every input is pinpointed to a unique output.
- Compute and elucidate the inverse of specified functions.
- Recognize and confirm the inverse connections present among functions.
Verified Answer
ZK
Zybrea Knight
May 02, 2024
Final Answer :
D
Explanation :
To show that two functions are inverse functions, we need to show that their composition in both orders is the identity function.
Using choice D, we have:
$f(g(x))=f(x^2-4)=\sqrt[7]{x^2-4+4}=x$
$g(f(x))=g(\sqrt[7]{x+4})=(\sqrt[7]{x+4})^2-4=x$
Therefore, $f(x)$ and $g(x)$ are inverse functions.
Using choice D, we have:
$f(g(x))=f(x^2-4)=\sqrt[7]{x^2-4+4}=x$
$g(f(x))=g(\sqrt[7]{x+4})=(\sqrt[7]{x+4})^2-4=x$
Therefore, $f(x)$ and $g(x)$ are inverse functions.
Learning Objectives
- Compute and elucidate the inverse of specified functions.
- Recognize and confirm the inverse connections present among functions.