Question Video: Determining the Monotonicity of a Function over an Interval | Nagwa Question Video: Determining the Monotonicity of a Function over an Interval | Nagwa

Question Video: Determining the Monotonicity of a Function over an Interval Mathematics • Second Year of Secondary School

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Which of the following is true of the function π(π₯) = π₯Β² β 1, where π₯ β€ 0? [A] π(π₯) is increasing on the interval (0, β). [B] π(π₯) is decreasing on the interval (0, β), and increasing on the interval (ββ, 0). [C] π(π₯) is decreasing on the interval (ββ, 0). [D] π(π₯) is increasing on the interval (0, β) and decreasing on the interval (ββ, 0). [E] π(π₯) is increasing on the interval (ββ, 0).

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Video Transcript

Which of the following is true of the function π of π₯ equals π₯ squared minus one, where π₯ is less than or equal to zero?

Well we have five options to choose from here. The first one option A is that π of π₯ is increasing on the interval zero, β, where neither end point is included in the interval; itβs an open interval. Option B is that π of π₯ is decreasing on the interval zero, β and increasing on the interval negative β, zero. Option C is that π of π₯ is decreasing on the interval negative β, zero. Option D is that π of π₯ is increasing on the interval zero, β and decreasing on the interval negative β, zero. And finally, option E is that π of π₯ is increasing on the interval negative β, zero.

All of these options have something to do with whether function π of π₯ is increasing or decreasing, and so this suggests that we should draw a graph. The function we have π of π₯ equals π₯ squared minus one is a quadratic function with constant term negative one. And so the π¦-intercept on the graph must be at π¦ equals negative one. What else is it helpful to know to draw this graph accurately? Well, itβs a quadratic function. So letβs find its root.

We can recognize π₯ squared minus one as the difference of two squares. Itβs π₯ plus one times π₯ minus one. And so its roots are negative one and one. So now we have three points on our graph, and we can draw a smooth curve β a parabola β going through those points. And we noticed that it is an upward facing curve, which makes sense given that the coefficient of π₯ squared is one, which is positive.

Okay, so are we ready to go through the options one by one yet? Well, not quite; thereβs also this bit here. It says βwhere π₯ is less than or equal to zero,β which defined the domain of our function. We therefore need to get rid of the part of the graph which is to the right of the π¦-axis.

Okay, now weβre ready. Letβs go through the options one by one. Is π of π₯ increasing on the interval zero, β? Well, either looking at the graph or the definition in the question, we see that actually π of π₯ isnβt even defined on this interval. So it canβt be increasing there. So this isnβt our correct option.

For a similar reason, we can rule out option B which says that π of π₯ is decreasing on the interval zero, β. Well, actually π of π₯ isnβt defined on that interval.

How about option C: π of π₯ is decreasing on the interval negative β, zero? Well, π of π₯ is at least defined on this interval. And if we go along the π₯-axis in the positive direction, we can see that the values of π of π₯ are getting smaller as π₯ increases. For example, the value of π of π₯ for this value of π₯ is less than the value of π₯ for a smaller value of π₯. As π₯ increases, π of π₯ decreases and this happens all the way to the end point, zero. This therefore is our answer.

Letβs have a look at the other two options that are left to make sure we can see why those are false. Option D says that π of π₯ is increasing on the interval zero, β. But of course, weβve said that π of π₯ is not defined there; itβs not part of the domain and so π of π₯ canβt be increasing there. So thatβs not true.

And finally, option E that π of π₯ is increasing on the interval negative β, zero. Well, π of π₯ is at least defined on this interval, but itβs decreasing on this interval as we saw earlier because as π₯ increases, the values of π of π₯ decrease. For itβs be increasing as π₯ increases, the values of π of π₯ would have to increase. And so this option E is also false.

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