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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

## Image Set Calculator

### 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 $a$.

### What is a domain of definition of a function?

The domain of definition of a function is the image set of all possible images by the function.

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