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In Exercises $45-60,$ for each function:a) Determine whether it is one-to-one.b) If the function is one-to-one, find a formula for the inverse.$$f(x)=2 x^{2}-x-1$$...

Question

In Exercises $45-60,$ for each function:a) Determine whether it is one-to-one.b) If the function is one-to-one, find a formula for the inverse.$$f(x)=2 x^{2}-x-1$$

In Exercises $45-60,$ for each function: a) Determine whether it is one-to-one. b) If the function is one-to-one, find a formula for the inverse. $$f(x)=2 x^{2}-x-1$$



Answers

In Exercises $45-60,$ for each function: a) Determine whether it is one-to-one. b) If the function is one-to-one, find a formula for the inverse. $$f(x)=2 x^{2}-x-1$$

This problem were given the function at the vaccine equals two X in the goldens problem. First goal is to determine this function is a 1 to 1 function. The easiest way to do this is to graph of function my graphing calculator. I'm gonna say that why equals two times X and plot that zoom out a little bit to get a better view of the entire function. So you see that this of a linear function with a slope of to know why intercepted zero. Now we need to perform a horizontal line test on this function. Or, in other words, if I was to draw a horizontal line through any section of death of axe, nor to pass the horizontal line test horizontal lines must only pass through the function once passed through more than once for any horizontal line fails the test. But we can see that with a linear function here were to draw a horizontal lines. It always dysfunction, always pass where it passes. The horizontal line test is any horizontal line drawn through another line, this case to acts on Lee intersex once, in other words, different inputs into the function f of X always have different or unique outputs. So that means that this function is what we call a one to one function that passed the horizontal line test so that we could find the function the interest function of F of X. First I piss ups to why, for half of Axel y equals two acts swamp X and Y or acts evils to. Why, in salt For why Why becomes acts divided by two? They're really substitute the inverse notation for why, or f minus one of acts F inverse of ax equals X over two. That is the expression for the interest function on the function after vaccine equals two X.

This problem in the function at the vaccine equals one divided by X in the first goal is problem. Problem is to determine the function. F is a 1 to 1 function. So the easiest way to do this is to graph the function and they perform the horizontal line test. So I'm gonna my graphing calculator in type in the function Why equals one divided my ex? I'll graph that function. You need to zoom in closer to the orange and to see this function more or less like this. So what we need to do now is we need to use the horizontal line test to see if this a 1 to 1 function or, in other words, so time to draw a horizontal line. Anywhere in this corn and plain does the horizontal line intersects the graph f of axe at one intersection point. I can see that I'm if I draw any horizontal line when they have your Y equals four y equals two the X axis itself. You know that's asking, told in this case or y equals negative, too. My horizontal lines only cross the graph of one point on the graph. One intersection So that is evidence that indeed this function passes horizontal line test hlt. That means that this function is indeed it one to one function or, in other words, and you. One unique input value for X corresponds to a unique output value. Now that we know it's a 1 to 1 function, it's science inverse function. I'm going to say that why equals one delighted by acts we need to swap accent. Why So acts equals one over Why, in salt for why a cross multiply, I get X y equals one sell for Why I'll get Why equals one over X this Friday and interest function notation. This is really interest function of f of ax equals one over X. This is sort of our case where the original function and inverse function are actually the same function. One of Rex but that is that is possible. We've seen that f of X passes the horizontal line test, so it is a watch one function and its inverse function is one over X as well

It is given FX equal toe hold by X If you draw the graph is a graphic. Something like this The graph something about this. So if you object with any whole gentle line, it's got well you once any hard interline. So this whole interline test that for using for gentle line best it is for one cool one. Therefore say this is it won't look for the car, the car Urgent matter This is the X axis This It'll y xs on this card apart language y equals before by X but kind of just I saw you How it's 11 now this right FX Equality poor by X So efforts say this is why we called for by ex not inverting X and y indigent x and y so you find this execrable four by your wife So finally y equals four by X Onda Finally, why isn't happiness x equal? Poor by X on This is one concert

Photo function affects is you could do express full. If the function affects is 11 then if fo phase you will do it will be then is must week will be. In that case, only FX will be 11 So we'll check. So if f off a is equal to a four b which means at a plus food is equal to be place for so by subtracting for on any side we get and we can get that age Akhil Toby So we can see that if F or physical do a forbidden aged also equal to be so we can say directly that FX is 11 No effect for fixes when we know we have to find involves function off this so and from love for in works So first off, all well supported. Why is equal toe little ways to go into effect? We can write it Why is it will do X plus four So exercise you will do Why minus four Who in worse off this will become effin worse off x No replace wide attacks this experience for So this is the inverse off


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