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How do you find the inverse function of #f(x) = (2x-3)/(x+4)#?

Answer 1

Cora Alexander

You can swap #x# with #y# and solve for #y#.

#y = f(x) = (2x - 3)/(x+4)#

#x = (2y - 3)/(y+4)#

#xy + 4x = 2y - 3#

#4x + 3 = 2y - xy#

#4x + 3 = (2 - x)y#

#y_"inv" = f_("inv")(x) = (4x + 3)/(2-x)#

#f(x) = (2x - 3)/(x+4)#: graph{y = (2x - 3)/(x+4) [-20, 20, -10, 10]}

#f_("inv")(x) = (4x + 3)/(2-x)#: graph{y = (4x + 3)/(2-x) [-20, 20, -10, 10]}

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Answer 2

Elijah Allen

To find the inverse function of ( f(x) = \frac{2x-3}{x+4} ):

Replace ( f(x) ) with ( y ).
Swap ( x ) and ( y ) to obtain ( x = \frac{2y - 3}{y + 4} ).
Solve this equation for ( y ).
First, multiply both sides by ( y + 4 ) to eliminate the denominator.
Then, distribute to get ( x(y + 4) = 2y - 3 ).
Expand: ( xy + 4x = 2y - 3 ).
Rearrange terms: ( xy - 2y = -4x - 3 ).
Factor out ( y ): ( y(x - 2) = -4x - 3 ).
Divide both sides by ( x - 2 ): ( y = \frac{-4x - 3}{x - 2} ).
Replace ( y ) with ( f^{-1}(x) ) to get the inverse function.

Therefore, the inverse function is ( f^{-1}(x) = \frac{-4x - 3}{x - 2} ).

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Answer from HIX Tutor

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