How do you verify inverses?
the question is y=3x-3...
we know how to make an inverse but how do you verify it using composition?
one way is to check to make sure that
f(f -1(x)) = x
and
f -1(f(x)) = x
For example, if
f(x) = 2x-7
then solve for x to get f -1
y = 2x-7
(y+7)/2 = x
so, f -1(x) = (x+7)/2
Now,
f -1(f(x)) = (f(x)+7)/2
= ((2x-7)+7)/2
= (2x)/2
= x
and
f(f -1(x)) = 2f -1(x) - 7
= 2*(x+7)/2 - 7
= x+7-7
= x
Using values, just plug 'em in:
f(3) = 6-7 = -1
f -1(-1) = (-1+7)/2 = 3
To verify the inverse of a function, you need to use composition. Here's a step-by-step guide on how to do it:
1. Start with the original function. In this case, the function is given as y = 3x - 3.
2. Replace y with the variable x and x with the variable y to create the inverse function. In this case, swapping x and y gives you x = 3y - 3.
3. Solve the inverse function for y. Rearrange the equation to isolate y: x + 3 = 3y. Divide both sides by 3 to get y = (x + 3)/3.
4. Now, you have the inverse function y = (x + 3)/3. This is the proposed inverse of the original function y = 3x - 3.
5. To verify that this is indeed the inverse, you need to perform a composition. To do this, take the original function and substitute in the inverse function, and then substitute the result back into the original function.
Composition with the original function:
y = 3x - 3
y = 3((x + 3)/3) - 3
6. Simplify the expression inside the parentheses:
y = (x + 3) - 3
y = x
7. Finally, you have arrived at the original input, which is x. Since y = x, this confirms that the proposed inverse function, y = (x + 3)/3, is indeed the correct inverse of the original function, y = 3x - 3.
By going through these steps and performing the composition, you can verify the inverse of a function.