# Verify that f(x)= (1-x^2) - (2+x) can be written as: -x + 2 - 3/(2+x)

• suegee3000
In summary, the conversation discusses how to verify that the equation f(x) = (1-x^2) - (2+x) can be written as -x + 2 - 3/(2+x). The individual has attempted different manipulations and factoring techniques, but is unable to find a solution. They are seeking guidance to steer them in the right direction. However, upon further examination, it is pointed out that the two sides of the equation are not equivalent for all values of x and therefore the initial statement cannot be proven.
suegee3000

## Homework Statement

verify that f(x)= (1-x^2) - (2+x) can be written as: -x + 2 - 3/(2+x)

## The Attempt at a Solution

i've tried manipulating as 1/(2+x) -x^2/(2+x), multiplying the orig. equation by (2-x)/(2-x), factoring, etc. nothing's worked, and i can't think of another approach. can anyone steer me in the right direction? thank you!

suegee3000 said:

## Homework Statement

verify that f(x)= (1-x^2) - (2+x) can be written as: -x + 2 - 3/(2+x)

## The Attempt at a Solution

i've tried manipulating as 1/(2+x) -x^2/(2+x), multiplying the orig. equation by (2-x)/(2-x), factoring, etc. nothing's worked, and i can't think of another approach. can anyone steer me in the right direction? thank you!

So basically you want to show:

$$(1-x^2) - (2+x) = -x + 2 - \frac{3}{2+x}$$

I'd be inclined to put the RHS over a common denominator, and then look to see what the next steps would be to distribution things out and start looking for simplifications...

On obvious point should be that the right hand side is NOT defined for x= -2 while the left hand side is. They can't possibly be equal! At first I thought that you mean "show they are equal for all x except -2" but then I notice that if x= 0, the left hand side is 1- 2= -1 while the right hand side is 1/2. And if x= 1, the left side is -3 while the right side is 0. Do you see my point?

## What is the given function and how can it be written in a different form?

The given function is f(x)= (1-x^2) - (2+x). It can be written in a different form as -x + 2 - 3/(2+x) by simplifying and rearranging the terms.

## What is the significance of writing the function in a different form?

Writing the function in a different form can make it easier to analyze and understand. It can also help in solving equations and finding the roots of the function.

## How do you verify that the two forms of the function are equivalent?

To verify that the two forms of the function are equivalent, we can use algebraic manipulation and substitution. We can show that both forms produce the same output for any given input value of x.

## What is the purpose of the variable x in the given function?

The variable x represents the independent variable in the function, which can take on different values. It is used to determine the output of the function.

## How can the given function be graphed and compared to its rewritten form?

The given function and its rewritten form can be graphed on a coordinate plane. We can then compare the two graphs to see that they are equivalent, with the same shape and intersecting points.

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