Sagot :
[tex]f\left( x \right) =\log _{ 2 }{ \left( x+4 \right) } \\ \\ \log _{ 2 }{ \left( x+4 \right) } =y\\ \\ { 2 }^{ y }=x+4[/tex]
[tex]\\ \\ x={ 2 }^{ y }-4\\ \\ \therefore \quad { f }^{ -1 }\left( x \right) ={ 2 }^{ x }-4\\ \\ \therefore \quad { f }^{ -1 }\left( 3 \right) ={ 2 }^{ 3 }-4=4[/tex]
[tex]\\ \\ x={ 2 }^{ y }-4\\ \\ \therefore \quad { f }^{ -1 }\left( x \right) ={ 2 }^{ x }-4\\ \\ \therefore \quad { f }^{ -1 }\left( 3 \right) ={ 2 }^{ 3 }-4=4[/tex]
The value of f^-1(3) is 4
What are inverse functions?
The inverse of a function f(x) is the opposite of the function
How to determine the inverse function?
The function f(x) is given as:
[tex]f(x) = log_2(x + 4)[/tex]
Express f(x) as y
[tex]y = log_2(x + 4)[/tex]
Swap the positions of x and y
[tex]x = log_2(y + 4)[/tex]
Express as exponents
[tex]2^x = y + 4[/tex]
Make y the subject
[tex]y = 2^x - 4[/tex]
Express the equations as an inverse function
[tex]f^{-1}(x) = 2^x - 4[/tex]
Substitute 3 for x in the above equation
[tex]f^{-1}(3) = 2^3 - 4[/tex]
Evaluate
[tex]f^{-1}(3) = 4[/tex]
Hence, the value of f^-1(3) is 4
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