### Video Transcript

What kind of function is depicted
in the graph? Is it (A) an even function, (B) a
logarithmic function, (C) a piecewise function, or (D) a polynomial function?

Let’s begin by providing a
definition for each of these terms. If a function 𝑓 of 𝑥 satisfies
the criteria 𝑓 of negative 𝑥 equals 𝑓 of 𝑥 for all 𝑥 within the domain of the
function, then it’s said to be even. We also know that these functions
have reflectional symmetry about the 𝑦-axis or the line 𝑥 equals zero. Then we look at logarithmic
functions. These are of the form log base 𝑎
of 𝑥. They’re the inverse to exponential
functions. It’s worth also noting that the
domain of a logarithmic function is the set of positive real numbers, and then the
range is the set of all real numbers.

Then we have piecewise
functions. And these are functions in which
more than one subfunction is used to define the output over different parts of the
domain. Each subfunction is then
individually defined over its own domain. Finally, we have polynomial
functions. Now, these are ones made up of the
sum or difference of constant terms, variables, and positive integer exponents such
as two 𝑥 cubed plus five 𝑥. The domain of a polynomial function
is the set of all real numbers. And we know that their graphs are
both continuous and smooth. In other words, there are no gaps
in the graph, which we might call a discontinuity, and there are no sharp corners on
the graph. So let’s look at our graph and
compare these definitions to it.

Firstly, we note that there is no
symmetry about the line 𝑥 equals zero. And so we see that the function
cannot be an even function. We also see that our graph is
certainly defined for values of 𝑥 greater than or equal to negative 10 and less
than or equal to eight. It might even be defined outside of
this interval, but we can’t be sure at this stage. What this does tell us is that the
domain is different to that of a logarithmic function, which is simply positive real
numbers. And so our graph cannot be the
graph of a logarithmic function.

And so we’re limited to piecewise
functions and polynomial functions. Now, in fact, we said that the
graph of a polynomial function is smooth; there are no sharp corners. And that means it’s differentiable
at every point. We can quite clearly see that our
graph has two sharp corners. And so it’s not smooth. Our graph cannot therefore be a
polynomial function. And so we’re left with (C) a
piecewise function. In fact, if we look carefully, we
see that there are three parts to this piecewise function. The first part is for values of 𝑥
less than negative three. We then have values of 𝑥 between
negative three and zero. And finally, our third subfunction
is values of 𝑥 greater than zero and certainly, from what we can see, up to
eight.