Well, I am wondering if it is possible to find inverses of parent functions?

Question

for constant = y=3
for identity = y=x
for quadratic = y=x^2
for cubic = y=x^3
for absolute value = y=|x|
for square root = y=√x
for reciprocal (odd power) = y=1/x
for reciprocal (even power) = y=1/x^2
for exponential (x>1) = y=10^x
for exponential (x<1) = y=10^x
for logarithmic (common) = y=log x
for logarithmic (natural) = y=ln x
and no, this isn't my homework. I know that Xs and Ys are supposed to switch...but then I don't know how to work from there. Do all these parent functions even have inverses (all)? Help please, I'm trying to understand something here. Work is appreciated!

Answer 1

The loose answer is, :Yeah, most of the time."

A function is a rule which takes 'things' from a collection X and 'pairs them up' with 'things' from a collection Y.

But inverses can be tricky. In some cases they simply don't exist and, in others, there may be restrictions. Indeed, there may be restrictions on the 'things' that a function can pair up. Let's look at your questions

y = 3 isn't a function (no matter what your math teacher may try to tell you) because it doesn't 'pair up' things from Y with anything. It simply specifies that Y contains a single element (y) that is paired up with 3. So, no inverse.

y = x is a function. It says that for every x in X there is one matching element y in Y. In this case it's kinda 'understood' that the sets X and Y are numbers, but it's also important to understand what *kinds* of numbers. If Y is the set of integers and X is the set of rationals, then it fails to be a function unless we 'restrict' the things from x to be only integers. If the X and Y sets are both integers, rationals, or reals, then it is a function and the inverse is x = y.

y = x² is a function (and again both X and Y must both be integers, rationals, or reals) The inverse is x = ±√y and this one has some *serious* restrictions. For openers, y can never be negative. And even if the X and Y are both integers or rationals, there will be values in Y whose square root won't exist in X.

In a loose sense, the 'inverse' function involves the 'inverse' operation.

Now it's your turn to do the rest of them so *you* understand it ☺


Doug

Answer 2

Doug thats an Odd answer

It is implied always that y = 3 means y(x) = 3 over some non empty domain for x.

A function is a 'rule' that maps one value to at most 1 outcome.

And yes every function has an inverse. If (x,f(x)) is a point mapped by f then (f(x),x) is a point mapped by the inverse of f.

The problem is that f^(-1) (the inverse) may not actually be a function itself.

A clear example of this is the inverse of f(x) = x^2

Then we could say that f^(-1)(x) = +/- sqrt(x) but this is clearly not a function since it maps 1 value to 2.

EDIT just to clarify i meant inverse in a general sense... but the traditional sense is that f(x) has an inverse f^(-1) if and only if f(f^(-1)(x)) = x and f^(-1)(f(x)) = x

In other words if both f and f^(-1) are 1-1. So for example you may consider f^(-1)(x) = =+/- sqrt (x) to be an inverse to f(x) = x^2 but strictly speaking it is not since f^(-1)(x) = =+/- sqrt (x) is not 1-1.

Thus the following functions have inverses from your list
y=x
y=x^3
y=sqrt(x)
y=1/x
y=10^x (both your cases... and to be honest why it is split up i have no idea)
y = log x
y = ln x

You are right in saying you need to 'switch them' and then rewrite it in terms of y again. If you can do it and you get a function (i.e., something that takes 1 value and gives you 1) then it is considered 'invertible'

Answer 3

yes there are inverses of any equation, what level of math are you in?

like inverse of absolute value is y= -|x|

parent functions are the basic form of all equations

look at the negative symbol, not as a negative, but the OPPOSITE of the number and it clears up a lot of different problems