How do you find the inverse of #h(x)= x^2 - 4x + 5# and is it a function?

Answer 1

Inversion gives two forms for #x = 2+-sqrt(h-1), h>=1 #

The for this equation is a parabola

#(x-2)^2=y-1>=0, and# so, #y>=1#,

The parabola is symmetrical about its axis x = 2.

Explicitly, the inverse functions are

#x = 2+-sqrt(h-1), h>=1 #, representing the halves of the parabola,

bifurcated by x = 2.l

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Answer 2

To find the inverse of a function, ( h(x) = x^2 - 4x + 5 ), follow these steps:

  1. Replace ( h(x) ) with ( y ).
  2. Swap ( x ) and ( y ), making the equation: ( x = y^2 - 4y + 5 ).
  3. Rearrange the equation to solve for ( y ).
  4. After solving for ( y ), replace ( y ) with ( f^{-1}(x) ), which represents the inverse function.

Whether the inverse function is also a function depends on whether the original function is one-to-one. A function is one-to-one if each input has a unique output. In this case, since the original function ( h(x) = x^2 - 4x + 5 ) is a quadratic, we can determine if it's one-to-one by checking if its graph passes the horizontal line test. If the graph intersects any horizontal line at most once, then the function is one-to-one, and its inverse will also be a function. If the graph intersects a horizontal line more than once, then it's not one-to-one, and its inverse won't be a function.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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