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Please help explain this problem thanks!
Remember that to find the inverse of a function, you are switching the x and y variables. So the inverse of this equation would be x=ay+b. If you isolate y, then the equation for the inverse becomes (x-b)/a=y. I use guess and check to find the answers.
I. Firstly, If a=-1, dividing by -1 is going to be the same as multiplying by negative one. So, if we multiply x by -1 in the first example, it is the same as dividing x by -1. So, we know that x is going to be the same in both the inverse equation and the normal equation. The b in f(x) is added to -x. In the inverse of f(x), the b is subtracted from -1, but IS ALSO DIVIDED BY -1, which means that the b is really just being added. Here is the example with numbers: Let a=-1. b=6, x=-2. -1(-2)+6=8 and (-2-6)/(-1)=8.
II. This one is easier because a and b are both given, so just do guess and check with a random x variable. But here’s the math behind it: dividing by 1 is the same as multiplying by 1. If a is 1 and b is zero, x is always just going to equal x in both equations because in f(x) it is multiplied by 1 and in inverse it is divided by 1 and nothing is added or subtracted from it. If we use the same x=-2 from the last problem -2=1(-2)+0 and inverse -2=(-2-0)/1
III. If we make a any real number and b is 0, the answer is always going to be different for the inverse and the regular function. Dividing x by a constant is different than multiplying x by that same constant for any real number that is not equal to 0 or 1. If we use guess and check and make a=4 and x=-2, we get f(x) -8=4(-2)+0 and inverse (-2+0)/4=-.5.
That’s why the answer is D
If any of this is confusing, please message me, and I would be glad to clarify.