Image of f(x) = x/(1+|x|): Find the Range

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In summary, the conversation discusses finding the image or range of a function f: R->R where f(x) = x/(1+|x|). The speaker is unsure but thinks the image may be R. They suggest splitting the function for positive and negative x values and using elementary theorems of analysis to determine the upper and lower bounds of the image. They also welcome any additional help in determining the image.
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Demonoid
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I've given a function where f: R->R, and need to determine an image(or range):

f(x) = x/(1+|x|)

I've pretty sure the image is R, but I'm not positive:

Heres my attempt:

y/1 = x/(1+|x|)
y(1+|x|) = x
y+ y|x| = x
y|x| = x - y... I'm kinda stuck here, since I can't determine an image from this ?


any help is welcome !
 
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  • #2
Split the function up in for positive and negative x. Then it should be easy to find upper and lower bounds. Together with some elementary theorems of analysis it should be easy to argue what the image can be.
 

FAQ: Image of f(x) = x/(1+|x|): Find the Range

What is the range of the function f(x) = x/(1+|x|)?

The range of the function f(x) = x/(1+|x|) is all real numbers except for 0. This means that the range includes all positive and negative numbers, but does not include 0.

How do I find the range of a function?

To find the range of a function, you can plug in different values for the input (x) and observe the corresponding output (y). Alternatively, you can use algebraic methods such as solving for y in terms of x or using the properties of the function to determine the range.

Can the range of a function be infinite?

Yes, the range of a function can be infinite if the function has an asymptote or if it has a continuous increasing or decreasing pattern with no upper or lower bound. In the case of f(x) = x/(1+|x|), the range is infinite as the function approaches positive or negative infinity as x approaches positive or negative infinity.

What is the domain of the function f(x) = x/(1+|x|)?

The domain of f(x) = x/(1+|x|) is all real numbers except for -1 and 1, as these values would result in a division by 0 which is undefined. This means that the function is defined for all values of x except -1 and 1.

How can I graph the function f(x) = x/(1+|x|)?

To graph the function f(x) = x/(1+|x|), you can plot points by choosing different values for x and calculating the corresponding values for y. You can also use a graphing calculator or software to plot the function. The graph will show a horizontal asymptote at y=1 and a vertical asymptote at x=0. Additionally, the graph will have a V-shaped curve with the point (0,0) as the vertex.

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