Find 1/f(x) for:
(a) f(x) = x^3 + 1
(b) f(x) = |x|
Thanks! In fact, we should go back to the basics.
Now, f(x) is read "f of x" Correct?
Just like so many students, we write f(x) all the time.
However, we do not have a clear view of f of x.
QUESTIONS:
(a) What exactly does f(x) really mean in math?
(b) Why does y = f(x) and vice-versa?
A function is a mapping between elements in one set D (the domain) and
another set C (the range), such that for every x in D there is one and only
one y in C.
We write f:x to y, and often f(x)=y, here f is a name for the function.
When we write something like f(x)=x^2+1 in elementary maths we are taking
as read that D is the subset of the reals R, for which the term on the right
of the equals sign makes sense and gives but a single value. Also we are
taking it as read that the range is that part of R which is the image under f
of D. This identifies the number in C which f maps x onto.
Now if we have a function f(x) when we talk about 1/f(x) we mean the
function g(x) such that for any x in D (and f(x)!=0):
g(x)=1/f(x).
so if f(x) = x^3 + 1, we have:
g(x)=1/f(x)=1/(x^3+1)
RonL
(a) What exactly does f(x) really mean in math?
f(x).
Here f is the function.
f(x) is the function when x is the, uh, what's the term here, input (?) or "thing" that the function f is "applied" onto.
f(3) is the function f "applied" on 3.
Say f is "the square of something minus 5"
If the something is x, then f(x) = x^2 -5
If the something is 3, then f(3) = 3^2 -5
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(b) Why does y = f(x) and vice-versa?
y = f(x) because we are used to the x,y rectangular or Cartesian axes. If x represents the domain of the function, then how do you graph the range f(x)? There is no f(x) in the said rectangular axes. So, we let y represent f(x). Hence let y = f(x) so that we may visualize the graph of f(x) on the x,y axes.
If the axes were (x,z), then let z = f(x) so that we may visualize the graph of f(x) on the x,z axes.
Vice-versa?
Not strictly. y may not equal f(x) always. But in the assumption or supposition y = f(x), since the two are supposed to be equal to each other, then the vice-versa is true.