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Determine whether or not the function determined by the given equation is one-to-one.$$g(x)=-4 x^{2}+1$$

No

Algebra

Chapter 4

Exponential and Logarithmic Functions

Section 1

Inverse Functions

Harvey Mudd College

University of Michigan - Ann Arbor

Idaho State University

Lectures

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So if we want to determine if this is one toe one or not, what we need to do is come up here and see if we plug into arbitrary numbers A and B. Um, and they have the same output. Then a has to be equal to be. So there's two ways you can kind of go about this the first way, which is the way I would probably recommend doing this. Um, it's just fine to numbers that you can plug into here to show that, um, they give the same output because we know this is a quadratic and quadratic normally look like a you, um So there has to be two values where this is equal. And and in this case, if you were to come and just play around with some numbers are actually think of where the axis of symmetry is, which is acceptable to zero in this case, because that's where Texas You can just plug in two numbers on either side and they should spend on the same value. So the easiest ones I would plug in is just like G a negative one, which that should output negative three. It looks like and then G of one is going to help put negative three a swell so notice these two are equal to each other, but negative one is not equal to one. And so then this implies that G of X is not want one, so that's the first way you could go about doing it. But if you want to be a little bit more rigorous or you have a hard time finding what these values are, you could first just go ahead and start from the definition. So doing G of A is equal to GDP and then seeing if you could get multiple outputs from this. So if we first go ahead and plug this in, that gives us negative for a square. Plus one is equal to negative for B squared plus one, and then we can subtract one divide by negative four, and I would give us a square. It is even to be squared now when we come over here to take the square root on either side, that's going to give us a is equal to plus or minus B, and so since a is just not one value, but it's multiple values here. That implies. So this implies G of X is not one toe one. So you let me just right way to here, so yeah, you can use either way. It doesn't really matter which way you go about it. It just depends on how rigorous you wanna be. Or essentially, how we easily you confined to values where these are actually equal to each other.

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