Find the inverse of each of the given functions.
[tex]f(x)=4 x-12[/tex]
[tex]h(x)=\frac{2 x-4}{3}[/tex]
[tex]f^{-1}(x)=\square x+\square[/tex]
[tex]h^{-1}(x)=\frac{3 x-12}{2}[/tex]
[tex]h^{-1}(x)=\frac{3}{(2 x-4)}[/tex]
[tex]h^{-1}(x)=\frac{3 x+4}{2}[/tex]
Answer (2)
To find the inverse of f ( x ) = 4 x − 12 , swap x and y and solve for y , resulting in f − 1 ( x ) = 4 1 x + 3 .
To find the inverse of h ( x ) = 3 2 x − 4 , swap x and y and solve for y , resulting in h − 1 ( x ) = 2 3 x + 4 .
Therefore, f − 1 ( x ) = 4 1 x + 3 .
And h − 1 ( x ) = 2 3 x + 4 .
Explanation
Problem Analysis We are given two functions, f ( x ) = 4 x − 12 and h ( x ) = f r a c 2 x − 4 3 , and we need to find their inverses, f − 1 ( x ) and h − 1 ( x ) .
Finding the Inverse of f(x) To find the inverse of f ( x ) = 4 x − 12 , we first replace f ( x ) with y , so we have y = 4 x − 12
Swapping x and y Next, we swap x and y :
x = 4 y − 12
Solving for y Now, we solve for y :
x + 12 = 4 y y = f r a c x + 12 4 y = f r a c 1 4 x + 3 So, f − 1 ( x ) = f r a c 1 4 x + 3 .
Finding the Inverse of h(x) To find the inverse of h ( x ) = f r a c 2 x − 4 3 , we first replace h ( x ) with y , so we have y = f r a c 2 x − 4 3
Swapping x and y Next, we swap x and y :
x = f r a c 2 y − 4 3
Solving for y Now, we solve for y :
3 x = 2 y − 4 3 x + 4 = 2 y y = f r a c 3 x + 4 2 So, h − 1 ( x ) = f r a c 3 x + 4 2 .
Final Answer Therefore, f − 1 ( x ) = f r a c 1 4 x + 3 and h − 1 ( x ) = f r a c 3 x + 4 2 .
Examples
In real life, inverse functions can be used to convert between different units of measurement. For example, if f ( x ) converts Celsius to Fahrenheit, then f − 1 ( x ) converts Fahrenheit back to Celsius. Understanding inverse functions helps in many practical conversion scenarios.
The inverse of the function f ( x ) = 4 x − 12 is f − 1 ( x ) = 4 1 x + 3 . The inverse of the function h ( x ) = 3 2 x − 4 is h − 1 ( x ) = 2 3 x + 4 .
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