Question

Help meeeeeeeeeeeeee

Answer 1

Step-by-step explanation:

Algorithm for deriving the formula of the inverse function

Step 1. In the formula for the original function, replace the notation of the argument and the value:

Step 2. From the resulting formula, express y(x).

Step 3. Take into account the constraints on the area of definition and the area of values of the original and/or inverse functions.

`\displaystyle\\ y=(1)/(2)x+2\\\\ 1.\ x=(1)/(2)y+2\\ \Rightarrow\ x-2=(1)/(2)y \\2.\ Multiply\ both\ parts\ of\ the\ equation\ by\ 2:\\\\2(x-2)=y\\\\2x-4=y\\Thus,\\y=2x-4\\eq 3x+(1)/(2) \\Answer:\ no\\\\2.\ y=5(x-2)\\\\1.\ x=5(y-2)\\\\2.\ Divide\ both\ parts\ of\ the\ equation \ by\ 5:\\\\(x)/(5) =y-2\\\\\Rightarrow\ (x)/(5)+2=y \\Thus,\\\\y=(x)/(5)+2 \\eq (1)/(5)(x+2)\\ Answer:\ no`

`y=2x-3\\1.\ x=2y-3\\\\2,\ x+3=2x\\\\Divide\ both\ parts\ of\ the \ equation\ by \ 2:\\\\(1)/(2)(x+3)=y\\\\ Thus,\\\\y=(1)/(2)x+(3)/(2) \equiv(1)/(2)x+(3)/(2) \\\\Answer:\ yes`

Answer 2

Answer: Choice D

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Step-by-step explanation:

To find the inverse, we swap x and y, then solve for y.

y = 2x-3

x = 2y-3

x+3 = 2y

2y = x+3

y = (x+3)/2

y = x/2 + 3/2

y = (1/2)x + 3/2

This shows the original function y = 2x-3 leads to the inverse y = (1/2)x+3/2, and vice versa.

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Another approach:

Since we're given a list of multiple choice answers, we can rule out the non-answers.

Plug x = 0 into the first equation of choice A. It leads to y = 2

That output 2 is then plugged into the second equation for choice A. It leads to y = 13/2. We do not get the original input 0, which shows that the equations in choice A do not undo one another. They aren't inverses of each other.

This allows us to rule out choice A. Choices B and C are similar stories.

Notice that plugging x = 0 into the first equation of choice D leads to the output y = -3. Then plug this as the input into y = (1/2)x+3/2 and you should get y = 0 to get us back where we started. This partially helps confirm we have a pair of functions that are inverses of each other. I'll let you try other values.