The function is one-to-one. Find its inverse_((x) = X-7' X770A 0"'(x)= ZZ,*+0 0 B. 0"'(x) = 77"f Oc f-1(x) = 7x+7 X+0~7+Tx 0 D. fF1()=...

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. $$f(x)=7-x$$

This problem, we're giving the function. Each of X equals seven minus sex. The first goals problems is to determine if the function age is a 1 to 1 function. The easiest way to do that to do this, to graph the function on a cornet plane and then before the horizontal line test. So I graph each of X here as y equals seven minus x. My graphing calculator. We'll zoom out a better picture of the function here. We could see that is simply a linear function with slope minus one in Why Intercept of seven. If I draw any horizontal line straight across the graph here, I can see that any horizontal line drawn across the linear function Onley intersects the line once. Whether that be in the X axis and sell for Y equals 50 or y equals negative. 50. Always horizontal lines only cross H of X once. Why intersection? That is evidence that age of acts passes. The horizontal line test passes the Hlt or his until lying test. And that means the Interfax is indeed a one toe one function, which means that for any unique and put value, there is a unique output value that corresponds to it. In other words, different inputs have different outputs. It's now that we know that h of X is 1 to 1 that's find this inverse function. We're gonna say that. Why? People seven minus X swap Excellent y to get X equals seven minus y Sol from why I'll get X minus seven equals minus. Why and the positive Why is equal to minus acts plus seven. All right and funk interest function notation To get interest function inch The perception of Age of X is evil of minus acts plus seven on down the line. Go ahead, circle that. So we assume that the original function H of X, given the problem isn't 1 to 1. Function in the US is in rece function. H inverters of X is evil that negative acts plus seven

This problem, we had the function G of X equals quantity X plus seven cute to the third power. The first goal This problem is to identify whether or not the function g a fax is a 1 to 1 function. The easiest way to do this is to graft the function and then perform the horizontal line test. So I wanted a graphing calculator and type in the function of Y equal in quantity X Plus seven. Most wanted me to a certain power. We can see that this fun claims a cubic cubic crunching translated or shifted to the left by seven units. And when I zoom in a little bit on the function so now the goal is to perform the horizontal line test so we can see a no matter where I drama arbitrary horizontal line to be down here, down here along the X axis cell for above decks access no matter where I draw a horizontal line where horizontal lines only cross the function at one intersection point even minus seven, it looks like it might cross. You might hit at many points she keeps doing in infinitely you were seeing while it is a close call. The horizontal line only intersects the graph right once for any horizontal life. This tells us that for any unique X input, there is a totally unique why output associated with it or different inputs along the X axis always have different. Why outputs? So that means that this function G affects indeed passes horizontal line test, which means that this function is indeed a one, So one function. So since the one that one function we can now solved within first function G Did you say that why equals X plus seven here swap extra wide to get X is equal to why plus seven to third power. No need to solve for why I'll take the cube root of both sides to get the queue grew of X equal y plus seven. It's a track seven from both sides to get wind equal, the cube root of acts minus seven or inverse function notation. This would be in first function. G of X is equal to the cube root of X minus seven. There's are inverse function formula because we know that the original function G of X is indeed a 1 to 1 function as it passes the horizontal line test

So for this problem we are asked to write the inverse of this function and I'm going to do that by changing ffx into an X. And acts into a Y. And then I'm gonna try to get this fly by itself. So the first thing I would do is at sea. Yeah. And then I would multiply both sides by B. Now to get rid of that cubed, I would take the cube root of both sides. Mhm. Mhm. I'm gonna go ahead and distribute that be mhm. And then my final step would be to add A. Which would be outside of the square root and that would be the inverse of the original function.

This problem, we have the function f of X equals X cubed plus seven. The first goal This problem is to determine at this function f of X. Is it 1 to 1 function? The easiest way to do this is to graft function. It performed the horizontal line test. It's only graphing calculator and then type in the function y equals X cubed plus seven. You see, there's a cubic function over a wide range X values, and the goal here is to see free pass a horizontal line in New Orleans. Graph. How many times with a horizontal line intersect the graph? We'll zoom in a little bit, so we did better view of that. Nearly horizontal part of the graph there around y equals seven that has a horizontal line at Y equal. Six would intersect the graph once or y equals 7.5. Intercept the graph once over here. If I goto y equals seven, it's close. Unify, Pass a line. The horizontal line through y angle. Seven. It only passes through the graph right along the y axis. Looks like it might be hitting several points along here, but actually, it's one point you keep zooming in zooming, and you see that more clearly so you can see a numerary pass horizontal line. Every horizontal line only passes through the graph once betweens that for any given ex employee value, there's only one unique wine value associated to it. So in that case, our function F does. Indeed, past passes the horizontal line test, which means that F of X is indeed in one to one function because they want to want to find it's inverse function. All right, why equals execute Plus seven. We need to swap X and y to get X equals. Why cute plus seven insult for Why X minus seven equals y Cubed Day two Q Group on both sides to get why equals Q Group of X, minus seven. All right, this is inverse function notation to be. The inverse function of F of X is equal to the Q Groot of X minus seven. There's our formula with inverse function of F effects, and we also know again F of X is a 1 to 1 function because it past the horizontal line test