Question Video: Determine and Analyze a Piecewise-Defined Function Mathematics

What kind of function is depicted in the graph? [A] an even function [B] a logarithmic function [C] a piecewise function [D] a polynomial function

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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.

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