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Image of a Function

Tool to calculate an image of a function. The image of a value z by the function f is the value of f(x) where x=z, also written f(z).

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Image of a Function -

Tag(s) : Functions, Geometry

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Image of a Function

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Tool to calculate an image of a function. The image of a value z by the function f is the value of f(x) where x=z, also written f(z).

Answers to Questions

What is the definition of an image by a function?

$ y $ is the image of $ x $ by the function $ f $ if $ f(x) = y $.

An image $ y $ exists if $ y $ belongs to the domain of definition of $ f $.

The image $ y $ by the function $ f $ is unique (there are never 2 images).

How to calculate an image of a function?

From the definition of the function

To find the image of a value $ a $ by a function $ f(x) $ whose formula/equation is known, is equivalent to compute $ f(x = a) = f(a) $.

Example: To calculate the image of $ 2 $ by the affine function $ f(x) = 3x + 1 $ is to compute $ 3 \times 2 + 1 = 7 $. So the image of $ 2 $ by $ f $ is $ f(2) = 7 $.

From the curve of the function

Finding the image of a value $ a $ by a function $ f $ whose curve is known, is to find the ordinate of the intersection of the curve with the abscissa line $ x = a $.

Example: Finding the image of $ 1 $ by the inverse function $ f(x) = 1/x $ is finding the intersection of the abscissa line $ x = 1 $ with the curve then go down to the corresponding ordinate: $ 1 $ so $ f(1) = 1 $.

What is the difference between image and preimage?

If a function $ f $ is such that $ f(x) = a $, the preimage of $ a $ by the function $ f $ is $ x $, and the image of $ x $ by the function $ f $ is $ a $.

Source code

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