### Video Transcript

Is the function π of π₯ equals
four π₯ minus three even, odd, or neither even nor odd?

So π of π₯ is an even function if
π of negative π₯ is equal to π of π₯, and π of π₯ is an odd function if π of
negative π₯ is equal to negative π of π₯. So notice that the only difference
between the definition of an even function and an odd function is this minus sign
just here.

So how do we find out if a function
is even or odd? We find what π of negative π₯
is. So what have I done here? Well Iβve taken the function π of
π₯ equals four π₯ minus three, and wherever Iβve seen an π₯, Iβve replaced it by a
negative π₯. So this π₯ here becomes a negative
π₯, and this π₯ here becomes a negative π₯.

So itβs basically exactly the same
procedure as when you substitute in a number to a function to evaluate it. Well now we can simplify this
function because four times negative π₯ is negative four π₯; we keep the minus
three.

Okay so now we can ask if π of π₯
is even. Well thatβs true if π of minus π₯
is equal to π of π₯. Well is negative four π₯ minus
three equal to four π₯ minus three? No itβs not.

Remember weβre not setting up an
equation where we have to solve for π₯. Weβre asking if these two functions
are the same. So does this hold for all values of
π₯? Sometimes we use an equal sign with
three lines to show this, that we need this to be true for all values of π₯ and not
just one or two.

This is not true for all values of
π₯. For example, itβs not true when π₯
is equal to one. When π₯ equals, one the left-hand
side is minus seven and the right-hand side is one. So these two functions cannot be
equal because they take the inputs one to different outputs.

So π of π₯ is not an even
function. If you show that a whole number is
not even, then that means that that whole number must be odd. But thatβs not true for functions;
a function can be neither even nor odd. So we have to check if π of π₯ is
odd. Itβs not guaranteed from the fact
that itβs not even.

How do we check out if π of π₯ is
odd? Well we compare π of negative π₯
to negative π of π₯ this time. So here, Iβve just used a
definition of π of π₯ and negated it to find the negative of π of π₯, and now we
can simplify.

And here weβve been careful to get
this plus here because weβre taking the negative of a bracket with a minus sign in,
and that becomes a plus. So this is a question that will
tell us if π of π₯ is odd. Is negative four π₯ plus three
equal to negative four π₯ minus three? And remember again, weβre talking
about this on the level of function, so are these identically equal for every value
of π₯? And yet again, the answer is
no. Try any value of π₯; it wonβt
work. And it has to be true for all of
those values of π₯, so this function is not odd.

So there we go, weβve seen this
function is neither even nor odd, and that fact might be surprising if youβre
thinking about whole numbers which are always either one or the other. So weβve shown this using the
definition, which involves π of negative π₯, but we could also have seen this by
drawing a graph. So letβs do that now.

Here is a very rough sketch of the
graph of the function π of π₯ equals four π₯ minus three; itβs a line with
π₯-intercept minus three and gradient four. So you might know that a function
is even if its graph has reflectional symmetry in the π¦-axis. And if you look here the graph,
youβll see that this graph clearly doesnβt have reflectional symmetry in the
π¦-axis. And so again, our conclusion is
that this function is not even.

A function is odd if its graph has
rotational symmetry, 180 degrees or order two, about the origin. So if you rotate that graph around
the origin by 180 degrees, do you get the same thing? Well no, and again our conclusion
is that it is not odd.

And so just as using the definition
involving π of negative π₯, we see by looking at the graph that the function π of
π₯ is neither even nor odd.