(a) Find the inverse of [tex]f(x)=3x-8[/tex]
(b) [tex]g(x)=\frac{x}{2}+7[/tex]
(c) [tex]h(x)=\frac{x-3}{x-2}[/tex]
Answer (2)
To find the inverse of f ( x ) = 3 x − 8 , swap x and y and solve for y , resulting in f − 1 ( x ) = f r a c x + 8 3 .
To find the inverse of g ( x ) = f r a c x 2 + 7 , swap x and y and solve for y , resulting in g − 1 ( x ) = 2 ( x − 7 ) .
To find the inverse of h ( x ) = f r a c x − 3 x − 2 , swap x and y and solve for y , resulting in h − 1 ( x ) = f r a c 2 x − 3 x − 1 .
The inverse functions are f − 1 ( x ) = f r a c x + 8 3 , g − 1 ( x ) = 2 ( x − 7 ) , and h − 1 ( x ) = f r a c 2 x − 3 x − 1 .
f − 1 ( x ) = 3 x + 8 , g − 1 ( x ) = 2 ( x − 7 ) , h − 1 ( x ) = x − 1 2 x − 3
Explanation
Problem Analysis We are given three functions and we need to find the inverse of each of them. The general strategy to find the inverse of a function is to replace f ( x ) with y , then swap x and y , and finally solve for y in terms of x . The resulting expression is the inverse function.
Finding the inverse of f(x) (a) Given f ( x ) = 3 x − 8 , we replace f ( x ) with y to get y = 3 x − 8 . Now, we swap x and y to get x = 3 y − 8 . We solve for y :
Add 8 to both sides: x + 8 = 3 y
Divide by 3: y = 3 x + 8
Therefore, the inverse function is f − 1 ( x ) = 3 x + 8 .
Finding the inverse of g(x) (b) Given g ( x ) = 2 x + 7 , we replace g ( x ) with y to get y = 2 x + 7 . Now, we swap x and y to get x = 2 y + 7 . We solve for y :
Subtract 7 from both sides: x − 7 = 2 y
Multiply by 2: y = 2 ( x − 7 )
Therefore, the inverse function is g − 1 ( x ) = 2 ( x − 7 ) .
Finding the inverse of h(x) (c) Given h ( x ) = x − 2 x − 3 , we replace h ( x ) with y to get y = x − 2 x − 3 . Now, we swap x and y to get x = y − 2 y − 3 . We solve for y :
Multiply both sides by ( y − 2 ) : x ( y − 2 ) = y − 3
Expand: x y − 2 x = y − 3
Rearrange to isolate y terms: x y − y = 2 x − 3
Factor out y : y ( x − 1 ) = 2 x − 3
Divide by ( x − 1 ) : y = x − 1 2 x − 3
Therefore, the inverse function is h − 1 ( x ) = x − 1 2 x − 3 .
Final Answer In summary, we found the inverse functions for the given functions:
f − 1 ( x ) = 3 x + 8
g − 1 ( x ) = 2 ( x − 7 )
h − 1 ( x ) = x − 1 2 x − 3
Examples
In real life, inverse functions can be used in various scenarios such as converting between temperature scales (Celsius and Fahrenheit), converting currencies, or decoding messages. For example, if f ( x ) converts Celsius to Fahrenheit, then f − 1 ( x ) converts Fahrenheit back to Celsius. Understanding inverse functions helps in reversing processes and solving for the original input given the output.
The inverse functions for the given functions are found by swapping variables and solving for the original input. The results are f − 1 ( x ) = 3 x + 8 , g − 1 ( x ) = 2 ( x − 7 ) , and h − 1 ( x ) = x − 1 2 x − 3 . This process involves systematically manipulating the equations to isolate the variable of interest.
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