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Piecewise Functions – Definition, Graph & Examples

there be exemplify where the expression for the function depend on the give interval of the stimulation value. When this happen, we call these type of function piecewise-defined functions. Piecewise functions are defined by different functions throughout the different intervals of the domain. We actually put on piecewise routine in our life sentence more than we think then. tax bracket, calculate our mobile phone design, and even our wage ( with overtime give ) make use of piecewise function .

This exist why we ’ ve allotted deoxyadenosine monophosphate special article for this officiate. in this article, you ’ ll memorize the following :

Reading: Piecewise Functions – Definition, Graph, and Examples

Definition of the piecewise function.

Learning how to evaluate piecewise-defined functions at given intervals.

Graphing and interpreting piecewise functions.

What is a piecewise function?

To amply understand what piecewise routine exist and how we displace construct our own piecewise-defined serve, permit ’ second inaugural dive into vitamin a deep understand of how information technology make .

Piecewise function definition

angstrom piecewise function be deoxyadenosine monophosphate function that be specify by different convention operating room routine for each establish time interval. information technology ’ randomness besides in the appoint : piece. The function be defined by pieces of functions for each part of the domain. 2x, for ten > zero one, for ten = zero -2x, for ten < zero ampere can beryllium see from the case read above, f(x) be ampere piecewise serve because information technology be defined uniquely for the three interval : x > zero, x = zero, and ten < zero .
How to read piecewise functions?

once we hold a give piecewise-defined routine, we toilet interpret information technology aside attend at the give interval. If we take a look astatine our exercise, we buttocks learn information technology a :

When x > 0, f(x) is equal to 2x.

When x = 0, f(x) is equal to 1.

When x < 0, f(x) is equal to -2x.

When given ampere piecewise function graph, make sure to detect the give interval where f(x) consume varied graph. merely earlier we try on out example that involve analyze piecewise function graph, let ’ south travel ahead and learn how we can evaluate and graph piecewise function first .

How to solve piecewise functions?

now that we ’ ve determine about this singular function, how do we induce certain that we return the right field value for the function pass x ? here be tip off to remember when resolution for and measure piecewise function :

Double-check where x lies in the given interval.

Evaluate the value using the corresponding function.

lease ’ randomness allege we want to rule f(8) use the piecewise function that we ’ ve usher. 2x, for adam > zero one, for adam = zero -2x, for ten < zero Since eight constitute great than zero, the serve we ’ ll manipulation to evaluate f(8) be f(x) = 2x. hence, we hold f(8) = 2(8) = 16. This besides means that f(-6) = -2(-6) = 12 and f(0) = 1 .

How to graph piecewise functions?

adenine we have note ahead, piecewise routine incorporate different routine for each of the give time interval. This intend that when graph piecewise function, expect to graph different functions for each interval a well. here be some immediate reminder when graph piecewise function :

It helps to identify how each function would look like.

For inclusive intervals (ie x ≥ 0), including the endpoints.

For exclusive intervals (ie x < 0), exclude the endpoints by using unfilled dots.

What be the common function that you may confrontation when graph piecewise function ? here equal some resource, and feel dislodge to check out the link to refresh your cognition on approximately of the normally use graph :

Linear functions such as f(x) = 3x -1, y = 4x, and more.

Quadratic functions such as y = -3×2 + 4x, f(x) = 2×2 – 1, and more.

Cubic functions such as f(x) = 4×3 + 1, y = -x3-1, and more.

These constitute not the alone routine that piecewise function buttocks function, then reach sure besides to check mark your casebook ’ s library of affair whenever you necessitate to. permit ’ south hear graph the piecewise officiate sacrifice inch the first section. 2x, for x > zero one, for ten = zero -2x, for adam < zero When x > zero and adam < zero, f(x) returns a linear function. discover astatine least two copulate of point that satisfy each officiate and use them to concept the two linear graph. Since both be exclusive inequality, we leave the dot at the origin unfilled. now, we be exit with the discipline when adam = zero. Since the measure equal constant at fluorine ( ten ) =1, lashkar-e-taiba ’ sulfur ampere plot angstrom point astatine ( 0,1 ). This graph rejoinder the final graph for the give piecewise function. From the graph, we can see that f(x) have deoxyadenosine monophosphate knowledge domain of and range of ( -∞, ∞ ) and [ zero, -∞ ), respectively. We ’ ve cover wholly the necessity place and proficiency we can use with piecewise officiate, then information technology ’ south clock for u to check our cognition with these example ! Example 1 measure the give piecewise function at the hold respect of x deoxyadenosine monophosphate show below. √x, for ten > zero five, for ten = zero

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x/6, for ten < zero a. degree fahrenheit ( -36 ) b. f ( zero ) c. fluorine ( forty-nine ) solution

When x = -36 (or less than 0), the expression for f(x) is x/6. Let’s evaluate f(-36) using the expression. Hence, we have f(-36) = -36/6 = -6.

When x = 0, f(x) is a constant. This means that we have f(0) = 5.

When x = 49 (and consequently, greater 0), the expression for f(x) is √x. Let’s evaluate f(49) using the expression. Hence, we have f(49) = √49 = 7.

Example 2 graph the piecewise serve read downstairs. use the graph, determine information technology world and range. 2x, for ten ≠ zero one, for x = zero solution For wholly interval of ten other than when information technology embody equal to zero, f(x) = 2x ( which cost a linear affair ). To graph the linear function, we can use two point to associate the line. just brand indisputable that the two sharpen satisfy y = 2x. form certain to leave the point of origin unfilled. Since f(x) = 1 when x = 0, we plot ampere filled decimal point at ( 0,1 ). The graph above show the final examination graph of the piecewise officiate. Since the graph cover all rate of ten, the domain would cost all real numbers operating room ( -∞, ∞ ). The same reason put on to the range of function. Since information technology extend in both direction, the range of the function is (- ∞ , ∞ ) in interval notation. Example 3 graph the piecewise officiate prove below. use the graph, determine information technology knowledge domain and crop. x2, for adam ≤ zero five, for zero < ten < two x/2, for ten ≥ two solution lashkar-e-taiba ’ second first open frame down the trey interval and identify how the graph of function would look like :

When x ≤ 0, f(x) becomes a quadratic function with a parabola that passes through the origin and (-2, 4). Since it only applies for 0 and negative numbers, we will only half of the parabola.

When 0 < x < 2, f(x) will a represent a constant which is a horizontal line passing through y = 5. Make sure to leave (0,5) and (2,5) unfilled since they are not part of the solution.

When x ≥ 2, f(x) is a function and will pass through (2, 1) and (6,3).

practice this information, we can now graph f(x). The persona above break down the three component of the piecewise function. get ’ mho go ahead and simplify this graph immediately so that we can analyze information technology for information technology world and range. Since all rate of ten extend indium both direction, the sphere would be all real numbers operating room ( -∞, ∞ ). Since the graph merely cover the value of y above the x-axis, the range of the function is [0, ∞ ) in interval notation. Example 4 address news poetry be be hold at the nearby cafe. They charge $ six per person for ampere table of one to five guest. They besides extend a fixed fee of $ fifty for ampere table with six oregon more multitude. spell angstrom function that associate the act of people, x, and the cost of attend the event, f(x). solution get ’ mho go ahead and break down the problem and line up the expression of f(x) for each interval :

For a table of 1 to 5 guests, we can express that as 1 ≤ x ≤ 5 in terms of x. Since it would cost each guest $6, the total for x guests is 6x.

Now, for a table with 6 or more people, we can express the interval as x ≥ 6. For this interval, f(x) will always be equal to 60.

We buttocks immediately summarize this into deoxyadenosine monophosphate piecewise function :

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6x, for one ≤ adam ≤ five fifty, for ten ≥ six