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Is the function 𝑓(π‘₯) = 4π‘₯ βˆ’ 3 even, odd, or neither even nor odd?

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Video Transcript

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

So 𝑓 of π‘₯ is an even function if 𝑓 of negative π‘₯ is equal to 𝑓 of π‘₯, and 𝑓 of π‘₯ is an odd function if 𝑓 of negative π‘₯ is equal to negative 𝑓 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 𝑓 of negative π‘₯ is. So what have I done here? Well I’ve taken the function 𝑓 of π‘₯ equals four π‘₯ minus three, and wherever I’ve seen an π‘₯, I’ve replaced it by a negative π‘₯. So this π‘₯ here becomes a negative π‘₯, and this π‘₯ here becomes a negative π‘₯.

So it’s 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 𝑓 of π‘₯ is even. Well that’s true if 𝑓 of minus π‘₯ is equal to 𝑓 of π‘₯. Well is negative four π‘₯ minus three equal to four π‘₯ minus three? No it’s not.

Remember we’re not setting up an equation where we have to solve for π‘₯. We’re 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, it’s 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 𝑓 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 that’s not true for functions; a function can be neither even nor odd. So we have to check if 𝑓 of π‘₯ is odd. It’s not guaranteed from the fact that it’s not even.

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

And here we’ve been careful to get this plus here because we’re 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 𝑓 of π‘₯ is odd. Is negative four π‘₯ plus three equal to negative four π‘₯ minus three? And remember again, we’re 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 won’t work. And it has to be true for all of those values of π‘₯, so this function is not odd.

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

Here is a very rough sketch of the graph of the function 𝑓 of π‘₯ equals four π‘₯ minus three; it’s 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 𝑦-axis. And if you look here the graph, you’ll see that this graph clearly doesn’t have reflectional symmetry in the 𝑦-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 𝑓 of negative π‘₯, we see by looking at the graph that the function 𝑓 of π‘₯ is neither even nor odd.

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