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).
Image of a Function - dCode
Tag(s) : Functions, Geometry
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$ 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).
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 $.
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 $.
The domain of definition of a function is the image set of all possible images by the function.