Question 274198
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Not "a variable like *[tex \Large x] or *[tex \Large y]"  This is function notation, like *[tex \Large f(x)].


You want to define *[tex \Large \phi(x)] such that *[tex \Large \phi(3)\ =\ 7] and *[tex \Large \phi(-1)\ =\ 5].


This is almost exactly the same thing as saying "What is an equation, in slope-intercept form, of the line that passes through the points (3, 7) and (-1, 5)?"


The only difference being that in one case you would derive an equation of the form *[tex \Large y\ =\ mx\ +\ b] but in the given case we will derive an equation of the form *[tex \Large \phi(x)\ =\ mx\ +\ b].


Use the two point form of the equation of a line:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ y\ -\ y_1\ =\ \left(\frac{y_1\ -\ y_2}{x_1\ -\ x_2}\right)(x\ -\ x_1) ]


Where *[tex \Large \left(x_1,y_1\right)] and *[tex \Large \left(x_2,y_2\right)] are the coordinates of the given points.


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ y\ -\ 7\ =\ \left(\frac{7\ -\ 5}{3\ -\ (-1)}\right)(x\ -\ 3)]


A little arithmetic:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ y\ -\ 7\ =\ \left(\frac{1}{2}\right)(x\ -\ 3)]


Distributing and collecting terms:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ y\ =\ \frac{1}{2}x\ -\ \frac{3}{2}\ +\ 7]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ y\ =\ \frac{1}{2}x\ +\ \frac{11}{2}]


And finally, replace *[tex \Large y] with *[tex \Large \phi(x)]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \phi(x)\ =\ \frac{1}{2}x\ +\ \frac{11}{2}]


John
*[tex \LARGE e^{i\pi} + 1 = 0]
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