What is the image of a function?

[luckyducky87]

Hi can someone please explain to me in simple english what the image of a function is... People have told me it is the range, however i have heard technically it is not the range but it is similar?

Thanks for your time cheers,
Lucky

 

[HallsofIvy]

First of all, it is not technically correct to talk about the image of a function. Strictly speaking we are talking about the image of a set under that fuction.

For example, the function $f(x)= x^2$ maps any number in the interval [-1, 1] into the interval [0, 1]. We say the "the image of [-1, 1] under the function $f(x)= x^2$ is [0, 1]. Sometimes you will see "the image of the function" to mean the image of the entire domain of the function f, under f. The "standard domain" for $f(x)= x^2$ is all real numbers which is mapped into the set of non-negative numbers, $[0, \infty)$ so we might say that the "image" of a function is its range but I do not consider that to be correct terminology. It is the image of the set, not the function.