Problems from Functions defined by parts

Indicate the domain and the image of the following function:

$f (x) = {\begin{array}{rcl} - 1 & if & x < - 1 \\ 2 x + 1 & if & - 1 \leq x < 2 \\ 2 & if & x \geq 3 \end{array}$

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Development:

We can find the domain of the first function from the intervals in its definition:

$D o m (f) = (- \infty, - 1) \cup [- 1, 2) \cup [3, + \infty) = (- \infty, 2) \cup [3, + \infty)$

To determine the image we can concentrate on the images of the different functions that compose the function, bearing in mind the domain where they are defined.

For $x < - 1$ or $x > 3$ we have no problems since we know the valuation of the function in these intervals.

For the straight line between $- 1$ and $2$ , we calculate the valuation in the above mentioned points:

$2 x + 1$ in $x = - 1$ values $- 1$

$2 x + 1$ in $x = 2$ values $5$

Therefore $I m (f) = [- 1, 5)$ .

It is necessary to bear in mind that we will include the extreme points in the image depending on whether they are included or not in the definition of the function.

Solution:

$D o m (f) = (- \infty, 2) \cup [3, + \infty)$ , $I m (f) = [- 1, 5)$

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