Inverse of a function (Gr 12 math)

[Senjai]

Inverse of a function.. (Gr 12 math) [SOLVED]

Homework Statement

If point (a,b) is on the line of y = f(x), what poin must be on the line of:

1. y = $$f\left(-\frac{1}{2}x\right)+1$$
2. *Trouble question* $$y=f^{-1}(x)+2$$

Homework Equations

The Attempt at a Solution

My answer for the first was : $$\left(-\frac{1}{2}a, b+2\right)$$

the second one bothers me... i know you have to swap the domain and range. but normally the questions i get are if y = 2x + 3, what is the inverse of the function. I then swap the x and y to get x = 2y + 3 then rearrange to end up with the answer y= (x-3)/2 but i don't really know where to start here...

I don't have an answer key for this worksheet. but what do i do, do i just go x = y +2 and do the reverse, then i would answer (a-2, b+2)? This is really bugging me.

Regards,
Senjai

 

[HallsofIvy]

Homework Statement

If point (a,b) is on the line of y = f(x), what poin must be on the line of:

1. y = $$f\left(-\frac{1}{2}x\right)+1$$
2. *Trouble question* $$y=f^{-1}(x)+2$$

Homework Equations

The Attempt at a Solution

My answer for the first was : $$\left(-\frac{1}{2}a, b+2\right)$$

No. You are completely misunderstanding the point of the problem. All you know about f is that f(a)= b so to be able to find f((1/2)x) at all, you must have (1/2)x= a. Now what is x? And, since f((1/2)x)= f(a)= b, what is y?

the second one bothers me... i know you have to swap the domain and range. but normally the questions i get are if y = 2x + 3, what is the inverse of the function. I then swap the x and y to get x = 2y + 3 then rearrange to end up with the answer y= (x-3)/2 but i don't really know where to start here...

I don't have an answer key for this worksheet. but what do i do, do i just go x = y +2 and do the reverse, then i would answer (a-2, b+2)? This is really bugging me.

What happened to f? This has nothing at all to do with "x=y+2" or "y= x=- 2".
Since all you know about f is that f(a)= b, all you know about f^-1 is that f^-1(b)= a. So in order to say anything at all about f^-1(x), you must have x= b. In that case, f^-1(x)= f^-1(b)= a. So what is f^-1(x)+ 2.

Notice that in neither of these questions are you asked anything very general- just to determine a single point on the graph.

Regards,
Senjai

 

[Architeuthis Dux]

Hello Senjai,

Thank you for sharing your thought process for solving the first problem. Your solution is correct. For the second problem, you are on the right track. The key is to remember that the inverse of a function is essentially "undoing" what the original function does. So, for the second problem, we can think of it as "undoing" the function y = f(x) + 2.

Starting with the original function, we can subtract 2 from both sides to isolate f(x) on one side:

y - 2 = f(x)

Then, we swap the x and y to get the inverse function:

x - 2 = f^{-1}(y)

Finally, we can rearrange to solve for y:

f^{-1}(y) = x - 2

Therefore, the inverse function of y = f(x) + 2 is y = x - 2. To find the point on this line that corresponds to the point (a,b) on the original line, we can simply substitute a for x and b for y:

y = x - 2

b = a - 2

Therefore, the point on the line y = f^{-1}(x) + 2 is (a, b-2).

I hope this helps clarify the process for finding the inverse of a function. Keep up the good work in your math studies!

Best,