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In this concept sheet, you will find the relevant information on the inverse of a first degree polynomial function (y = ax + b).
To graphically determine the inverse of a linear function, proceed as follows:
Sketch the linear function of which we want the inverse.
Sketch the axis of symmetry |y = x|.
Perform a reflection on the linear function with respect to the line. |y = x|.
Sketch the inverse of the following linear function: |y = 3x - 6|.
Sketch the initial function.
Sketch the axis of symmetry |y = x|.
Perform a reflection (in red) of the initial linear function (in blue) with respect to the axis of symmetry (in black).
In the GeoGebra animation above, you can move the y-intercept and the x-intercept of the blue line to see what is happening. It is also possible to invert the coordinates of certain points.
If a function has the points ( 6 , 9 ), ( 9 , 2 ), and ( 11 , -2 ), the inverse of the function will have the following points : ( 9 , 6 ), ( 2 , 9 ), and ( -2 , 11 ).
To determine the inverse of a linear function algebraically, proceed as follows:
In the linear function rule, invert the variables |x| and |y|.
Isolate the variable |y|.
Determine the inverse rule of the following function algebraically:
|y = 3x - 1|
Swap the variables |x| and |y|.
|x = 3y - 1|
Isolate the variable |y|.
|x \color{red}{+ 1} = 3y - 1 \color{red}{+ 1}|
|x + 1 = 3y|
|\displaystyle \frac{(x + 1)}{\color{red}{3}} = \frac{3y}{\color{red}{3}}|
|\displaystyle \frac{x + 1}{3} = y|
Answer:
The inverse of the initial function is: |\displaystyle y = \frac{x + 1}{3}|.