Question regarding inverse functions

In summary, the inverse of f(x+3) is not equal to f^{-1}(x+3) for all invertible functions, as shown in the counterexample provided. The inverse of a function is found by switching the x and y variables and solving for y.
  • #1
michellemich
1
0
f(x) where x belongs to all real numbers
inverse: f-1(x), where x belongs to all real numbers

True or False:
The inverse of f(x+3) is f-1(x+3)

My ideas:
I think that it is false given that when you usually find the inverse of a function, you switch the x and y variables and solve for y again meaning that the inverse couldn't stay the same.
I figured since the domain and range of f(x) belong to all real numbers, possibly f(x) = x and then inputting f(x+3) = x+ 3
then y = x+3
then y = x - 3 but I am not really sure if that's right :s
 
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  • #2
michellemich said:
f(x) where x belongs to all real numbers
inverse: f-1(x), where x belongs to all real numbers

True or False:
The inverse of f(x+3) is f-1(x+3)

My ideas:
I think that it is false given that when you usually find the inverse of a function, you switch the x and y variables and solve for y again meaning that the inverse couldn't stay the same.
I figured since the domain and range of f(x) belong to all real numbers, possibly f(x) = x and then inputting f(x+3) = x+ 3
then y = x+3
then y = x - 3 but I am not really sure if that's right :s

You are given that ##f## has an inverse ##f^{-1}##. What happens when you solve the equation ##y=f(x+3)## for ##x##?
 
  • #3
Good Day michellemich!

If you are not sure of your answer, try some composition: let your original function be f(x)and your questionable inverse function be g(x)

Evaluate (f of g) and (g of f). If they undo each other, they are inverses.
 
  • #4
If you want to know if this is true for all invertible functions, it is simple enough to find a counterexample.

If, say, f(x)= 2x+ 3, then [itex]f(x)= 3x- 2[/itex], then [itex]f^{-1}(x)= (x+ 2)/3[/itex]. f(x+3)= 3(x+ 3)- 2= 3x+ 7. The inverse of that function is [itex](x- 7)/3[/itex]. Is that equal to [itex]f^{-1}(x+ 3)= (x+3+ 2)/3= (x+ 5)/3[/itex]?
 
  • #5


Your thinking is correct. The statement is false. The inverse of f(x+3) would be f-1(x+3), not f-1(x). This is because the function f(x+3) is essentially f(x) shifted 3 units to the left. Therefore, the inverse would also be shifted 3 units to the left. In general, when dealing with inverse functions, you cannot simply add or subtract numbers to the function and expect the inverse to remain the same. The variables must be switched and the inverse must be solved for again.
 

What is an inverse function?

An inverse function is a function that undoes the action of another function. In other words, if the original function takes an input and produces an output, the inverse function takes that output and produces the original input.

How do you find the inverse of a function?

To find the inverse of a function, you need to switch the input and output variables and solve for the new output variable. This can be done algebraically by isolating the output variable on one side of the equation, or graphically by reflecting the points of the original function over the line y=x.

Can all functions have an inverse?

No, not all functions have an inverse. A function must be one-to-one (each input has only one output) for it to have an inverse. If a function fails the horizontal line test, it does not have an inverse.

What is the notation for inverse functions?

The notation for inverse functions is f-1(x). This does not mean 1/f(x), but rather the inverse of the function f(x).

What is the domain and range of an inverse function?

The domain of an inverse function is the range of the original function, and the range of an inverse function is the domain of the original function. In other words, the inputs and outputs of the original function become the outputs and inputs of the inverse function, respectively.

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