how to get domain and range of piecewise function

A piecewise-defined function is one that is described non by a uncomparable (single) equation, but by two operating room more. Allow the following function definition:

\[F(x) = \left\{\begin{matrix}-2x, -1 \leq x < 0\\X^{2}, 0 \leq x < 1 \end{matrix}\right.\]

Above mentioned piecewise par is an case of an equation for piecewise function delimited, which states that the function definition is divers on antithetical parts of its world. For the piecewise-characterised function in a higher place, the domain is [−1, 1][−1, 1], but the function definition on [−1, 0][−1, 0] is distinct from that of function definition connected [0, 1][0, 1].

How to Uncovering the Domain of a Piecewise Function

Let's learn to find the domain and range of the piecewise run

Consider the operate: y = x² if x < 0, y = x + 2 if 0 ≤ x ≤ 3, y = 4 if x > 3?

World: (−∞, ∞)

Range: (0, ∞)

Result:

It is ideal to begin graphing piecewise functions by first thoroughly recitation the "if" statements and you will then perhaps shorten the chance of making an error past doing sol.

Having said that, we have:

\[\left\{\begin{matrix} y = x^{2} \text{ if x < 0} \\ y = x + 2 \text{ if 0 }\leq x \leq 3 \\ y = 4 \text{ if x }> 3\end{intercellular substance}\right.\]

It is quite crucial to consider the greater/less than or isochronous to signs because two points along the same domain wish seduce IT such that the graph is not a subroutine. Nonetheless:

When No Restrictions in the 'if' Statements

y = x² is a easy parabola, and you believably get laid that it begins at the origin, (0, 0), and stretches out indefinitely in both directions. Merely, our limitation is entirely x-values little than (<) 0, thus we will only draw the left uncomplete of the graphical record, and leave an open circle at the point (0, 0), as the limitation is less than 0, and does not admit 0.

Our next graph is a normal simple mathematical function moved upwards by two but only appears from 0 to 3 and includes both, so we leave draw the graph from 0 to 3, with shaded circles on some 0 and 3.

The last-ditch social function is the simplest function, a constant function of y = 4, where there is only a horizontal line at the value of 4 on the y-Axis, but only when aft 3 on the x-axis of rotation, because of our limitation.

Let's see how it would appear without the limitation:

(Image will be uploaded soon)

When Adding Restrictions in the 'if' Statements

At once, let's find the world and kitchen range of a piecewise function adding the restrictions in the 'if' statements:

Similar we said earlier, the quadratic equation just looks the likes of less than cipher (<0), the linear only looks like from 0 to 3, and the constant exclusively appears followed past 3, thus:

Domain: (−∞, ∞)

Range: (0, ∞)

Our domain is all real numbers game because of our x-values being continuous along the x-axis, seeing that we have unity shaded ring on the linear function at x = 0, and one shaded circle on the linear function at x = 3. The constant function continues endlessly to the right thus, despite the functions visually fillet, the graph still continues, therefore, all real numbers.

Our range begins at 0, simply doesn't include it, and goes until infinity because of the graph not active below y = 0, and the lowest compass point being the quadratic not touching the x-axis at the origin, (0, 0), and stretches retired unendingly upwards.

(Image will be uploaded soon)

Solved Examples

Resolution piecewise functions requires plotting graphs. Let's understand how to deal with a piecewise-defined function

Example:

Consider the function described as follows.

\[\left wing\{\begin{matrix} y = x + 2 \text{ if }x < 0 \\ 2 \text{ for }0 \leq x \leq 1 \\ -x + 3 \schoolbook{ for }x > 1\end{ground substance}\right.\]

Solution:

In this example, the routine is piecewise-linear, since apiece of the tierce parts of the graph is a line.

Piecewise-defined functions tush also incorporate discontinuities ("jumps"). The role in the example down the stairs consist of discontinuities at x = −2x = −2 and x = 2.

Example:

Graph the function described as given below:

\[\left\{\begin{matrix} y = 1/2x^{2} \text{ if }x < -2 \\ 0 \text{ for }-2 \leq x < 2 \\ 1/2x^{2} \schoolbook{ for }x \leq 2\closing{matrix}\right.\]

Note that we withdraw the assistant of Pieris rapae circles in the graph in order to indicate that the end point of a curve is non included in the graph, and solid dots to evidenc endpoints that are included.

Example:

Graph the function defined below.

y = logx for 0<x<1

1/(x−2) for x≥1

Solution:

Negative values of x and 0 are excluded in the domain since the 1 st function, logx, is not defined for those values. The value x=2 is non enclosed in the domain seeing that the 2 nd social occasion is undefined for that valuate (it contains a erectile asymptote there). Thus, the domain of this use is {x | 0<x<2}∪{x | x>2}. This can be illustrated using interval notation as (0,2)∪(2,∞).

how to get domain and range of piecewise function

Source: https://www.vedantu.com/maths/piecewise-functions