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Consider the function defined by$$y=f(x)=\left\{\begin{array}{cc}x & 0 \leq x < 1 \\ -x & x \geq 1\end{array}\right.$$sketch the graph of this function and determine if it is one-to-one.

It is one-to-one

Algebra

Chapter 4

Exponential and Logarithmic Functions

Section 1

Inverse Functions

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So first, let's go ahead and sketch this graph here and then Once we have that sketched, we can apply the horizontal line test to see if this is going to be 1 to 1 or not. So, um, this starts from X 0 to 0 to one. So actually, I only need the first quadrant. I don't need the rest of this here. Then I'll just make this small bit bigger. Okay, so let's put down some tick marks. So first, this first part, it says it's just X, um, from 0 to 1. So zero is included. So if I plug in zero, that was just giving me zero. If I plug in one, I guess we won. But I'm excluding this. It would be an open circle, and then we know two lines. We just connect the end points like that. Then for the second one, we're going to go ahead and plug in one first, so that would be negative one. So, actually, let me put some tick marks going down as well around there, and we're going to include it this time. So we have a closed circle, and then if we plug in like two. That would be negative, too. And then we would just go ahead and connect it like that, and then it keeps on going forever out. So this is the graph of that piecewise function and notice that this does pass the horizontal line test because if we just draw a bunch of horizontal lines, we only ever cross them once. Let me get rid of those. So yes, 1 to 1 since passes the horizontal wine just and actually keep this one in mind for one of the later problems. About if there is a function that is, or if all 1 to 1 functions have to be increasing or decreasing, because this is actually a counter example to that one you'll see later on.

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