# Solve Image of f(x)=-2(x+1)/(x-1)^2 in x from [0,1)U(1,3]

• MHB
• Vali
In summary, the function f(x)=-2(x+1)/(x-1)^2 with x in the interval [0,1)U(1,3] has an image of (-infinity, -2]. To find this, one can use the derivative f'(x)=2(x+3)/(x-1)^3 and observe that f'(x)<0 when x is in the interval [0,1) and f'(x)>0 when x is in the interval (1,3]. This means that f(x) is decreasing on [0,1) and increasing on (1,3], with f(0)=f(3)=-2 and f(x) approaching -infinity as x approaches 1
Vali
I have the next function:
f(x)= -2(x+1)/(x-1)^2
x is from [0,1)U(1,3]
I need to find the image of the function which is (-infinity, -2].I tried with the derivative but when I solve f'(x)=0 I obtain x=-3 which is not from x interval.I don't know how to continue.

I presume you got
$$f'(x)\ =\ \frac{2(x+3)}{(x-1)^3}$$
which is correct. Notice that $f'(x)<0$ when $x\in[0,\,1)$ and $f'(x)>0$ when $x\in(1,\,3]$. So $f(x)$ is decreasing on $[0,\,1)$ and increasing on $(1,\,3]$. Also, $f(0)=f(3)=-2$ and $f(x)\to-\infty$ as $x$ tends to $1$ from either side. From these, you should be able to piece together the range of $f(x)$.

Last edited:
Thanks a lot!

## 1. What is the domain of the function f(x)=-2(x+1)/(x-1)^2?

The domain of this function is [0,1)U(1,3). This means that the function is defined for all values of x between 0 and 1, including 0, and for all values of x between 1 and 3, but not including 3.

## 2. What is the range of the function f(x)=-2(x+1)/(x-1)^2?

The range of this function is all real numbers except for -2. This is because the function is undefined at x=1 and approaches -2 as x approaches either 0 or 3.

## 3. What is the behavior of the function as x approaches 1?

As x approaches 1, the function approaches infinity. This is because the denominator of the function becomes very close to 0, making the value of the function very large.

## 4. What are the critical points of the function f(x)=-2(x+1)/(x-1)^2?

The critical points of this function are x=0 and x=3. These are the points where the function changes direction and the slope of the graph is 0.

## 5. How can I graph the function f(x)=-2(x+1)/(x-1)^2?

To graph this function, you can plot the critical points at x=0 and x=3, and then use the behavior of the function at these points to determine the shape of the graph. You can also plot a few additional points to get a better understanding of the graph. Alternatively, you can use a graphing calculator or software to graph the function quickly and accurately.

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