inverse functions and checking them

[kangsang24]

The problem asked to find the inverse of y=-sqrt(1-x) then check to see if it is right by composing f (x) with f-(x).
Know the inverse is 1-X^2. and I plugged it back in to get -sqrt (x). Which my teacher said is x because absolutrd value of x when x <0 is -x. Which I cant wrap my head around!

 

[richardt]

Greetings: Given f(x) = -sqrt(1-x), it follows that f^-1(x) = 1 - x² where x < or = 0 [f(x) has range y < or = 0; the domain of f^-1 is equal to the range of f].

Now, f(f^-1(x)) = -sqrt[1 - (1 - x²)] = -sqrt[x²]. I believe your problem is in not recognizing that sqrt(x²) = |x| not x. For instance, sqrt[(-5)²] = 5 = |-5|. That said,

f(f^-1(x)) = -|x|. But since the domain of f^-1 < or = 0, |x| = -x. [e.g., if x = -3, then |x| = 3 = -x].

Therefore, f(f^-1(x)) = -|x| = -(-x) = x.

I hope this helps.

Rich

 

[Quaid]

my teacher said ... absolute value of x when x<0 is -x. Which I cant wrap my head around

Remember that the notation -x means (-1)(x)

When x itself is negative, multiplying it by -1 yields a positive result.

That's what you want, for the absolute value.


EG:

X = -4

The absolute value of -4 is 4 (you know that)

(-1)(-4) = 4

See that? We just multiplied X by -1 to change a negative value into its opposite.

This is what we do when taking the absolute value of any negative number.

That is why |X| = -X when X itself is negative.

When X is not negative, then X is the same as its absolute value.

That is why |X| = X when X is non-negative.

Cheers :cool: