Question 537893
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First some terminology:  *[tex \Large f] is a typical way of representing a function.  A function is a mathematical relationship between one or more independent variables (the <i><b>input</b></i>) and a dependent variable (the <i><b>output</b></i>).  For now, we will only consider functions that have a single input or independent variable.  A function must have the property that for any given independent variable, you can only have a single dependent variable value.


Think of a mathematical relationship like a box.  On the front end of the box is a slot, like a coin slot on a gumball machine, and on the other end is a tray, like where you get the gumball once you have turned the knob on the machine.  And speaking of the knob or crank, that is on the side of the box.  You take an input value, or independent variable value, or <i><b>x</b></i> and insert it into the slot in the front.  Then you turn the crank and an independent variable value (*[tex \Large y\ =\ f(x)]) lands in the tray on the other end.


Now if the machinery inside the box allows different values of *[tex \Large f(x)] to be generated when some particular value of *[tex \Large x] is put into the slot, for example if you put *[tex \Large 4] in the slot and sometimes you get *[tex \Large 2] out in the tray but sometimes you get *[tex \Large -2] instead, then this particular machine IS NOT a function, it is simply a relation.


Note that it is perfectly ok for the machine to give you the same value for different values of the input, for example if I put in the number *[tex \Large \frac{\pi}{2}] and get *[tex \Large 1] as an output and then put in the number *[tex \Large \frac{3\pi}{2}] and also get *[tex \Large 1] out, then we still have a function.


As I mentioned earlier, *[tex \Large f] is a typical way to represent a function.  This needs to be differentiated from *[tex \Large f(x)] which specifically means "the value of the function *[tex \Large f] at *[tex \Large x]"


In most cases when you see something like *[tex \Large f(x)\ =\ 4x\ -\ 5], this is just a definition of the inner workings of one particular machine that has a slot, a tray, and a crank.


Your problem is asking you to describe what happens when, instead of *[tex \Large x] being submitted to the machine, you submit *[tex \Large 3a\ +\ 1].  The answer, in general, is that whatever was supposed to happen to *[tex \Large x] must now happen to *[tex \Large 3a\ +\ 1].  The process is quite simple:  Take the definition of your function:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ f(x)\ =\ 4x\ -\ 5]


And every place you see an *[tex \Large x], replace it with *[tex \Large 3a\ +\ 1], thus:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ f(3a\ +\ 1)\ =\ 4(3a\ +\ 1)\ -\ 5]


The only thing left to do is to use the distributive property and collect like terms in the RHS to simplify.


<i><b>Super-deluxe Double-plus Extra Credit</b></i>


Write the meaning of *[tex \LARGE f(3a\ +\ 1)] in an English sentence.


John
*[tex \LARGE e^{i\pi} + 1 = 0]
My calculator said it, I believe it, that settles it
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