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In this lesson, weβll learn how to identify, write, and evaluate a piecewise function given both the function equation and the graph of the function.
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Letβs begin with a definition.
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A piecewise function is a function thatβs made up from pieces of more than one different function.
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And each part of the function is defined on a given interval.
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For example, letβs imagine we have the function π of π₯, and itβs a piecewise function defined by π₯ plus one when π₯ is less than three and two π₯ minus two if π₯ is greater than or equal to three.
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In other words, for all values of π₯ up to but not including π₯ equals three, we would use the function π of π₯ equals π₯ plus one.
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Then, when π₯ is equal to and greater than three, we use the function two π₯ minus two.
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And then if we wanted to evaluate the function at a specific value of π₯, we need to be careful to follow these rules.
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We can also sketch the graph of this piecewise function.
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Up to but not including π₯ equals three, we use the function π of π₯ equals π₯ plus one.
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The graph of this function looks as shown.
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Notice that Iβve included an empty dot at π₯ equals three, and thatβs because the function isnβt defined by π of π₯ equals π₯ plus one here.
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It is, however, defined at π₯ equals three but by the function two π₯ minus two.
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And so the graph could look a little something like this.
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We could also include a solid dot at π₯ equals three to show that the function is defined here if we chose.
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Letβs have a look at an example of how to evaluate a piecewise function at a given value of π₯.
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Given that the function π of π₯ is equal to six π₯ minus two if π₯ is less than negative six, negative nine π₯ squared minus one if π₯ is greater than or equal to negative six and less than or equal to eight, and negative five π₯ cubed plus four if π₯ is greater than eight, find the value of π of four.
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We see that π of π₯ is a piecewise function, and itβs defined by three separate functions.
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When π₯ is less than negative six, weβre going to use the function π of π₯ equals six π₯ minus two.
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When π₯ is between negative six and eight and including those values, we use the function negative nine π₯ squared minus one.
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And when π₯ is greater than eight, we use the function π of π₯ is negative five π₯ cubed plus four.
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Now we want to find the value of π of four.
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And so we need to make sure that we correctly select the function that we need to use when π₯ is equal to four.
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Well, four is between negative six and eight.
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So weβre going to use this part of the function: π of π₯ is negative nine π₯ squared minus one.
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And so, π of four is found by substituting π₯ equals four into this function.
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Itβs negative nine times four squared minus one.
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Now, of course, the order of operations, which is sometimes abbreviated to PEMDAS or BIDMAS, tells us to begin by working out the value of the number being raised to some exponent.
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So in this case, we begin by working out four squared.
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Thatβs four times four which is 16.
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And so our calculation becomes negative nine times 16 minus one.
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We then perform the multiplication part of this calculation, remembering that a negative multiplied by a positive is a negative.
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We get negative 144 minus one.
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Negative 144 minus one is negative 145.
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And so, given the piecewise function π of π₯, we see that π of four is negative 145.
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Weβre now going to look at how to apply this process but when working with composite functions based off of an individual piecewise function.
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Consider the function π of π₯ is equal to π₯ plus four if π₯ is greater than four, two π₯ if π₯ is greater than or equal to negative one and less than or equal to four, and negative three if π₯ is less than negative one.
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Find π of π of two.
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π of π of two is a composite function.
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Itβs a function of a function.
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Weβre going to begin by looking at the inner function first, so weβre going to begin by thinking about π of two.
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Now, our π of π₯ is a piecewise function, and itβs defined by different functions on different intervals of π₯.
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Weβre told that when π₯ is greater than four to use the function π₯ plus four.
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When π₯ is between and including negative one and four, we use the function two π₯.
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And when π₯ is less than negative one, we use the function π of π₯ equals negative three.
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Two, of course, lies between negative one and four, and so weβre going to use the second part of our function.
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That is, when π₯ is equal to two, π of π₯ is equal to the function two π₯.
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And so, π of two is found by substituting two into this equation.
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We get two times two, which is four.
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So we found π of two; itβs four.
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If we replace π of two with its value of four, we see that we now need to evaluate π of four.
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And we need to be really careful here.
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Weβre actually still using this second part of the function.
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And this is because we only use the first part of the function when π₯ is strictly greater than four.
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When itβs less than or equal to four, we use the function two π₯.
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And so once again, we substitute our value of π₯ into the function π of π₯ equals two π₯, so itβs two times four which is equal to eight.
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Given our piecewise function, π of π of two is eight.
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In our next example, weβll see how to complete a table of values for a piecewise function.
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Find the missing table values for the piecewise function π of π₯, which is equal to two to the power of π₯ if π₯ is less than negative two, three to the power of π₯ if π₯ is greater than or equal to negative two and less than three, or two to the power of π₯ if π₯ is greater than or equal to three.
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And then we have a table with the values of π₯, negative three, zero, and three.
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Remember, when we have a function defined by different functions depending on its value of π₯, we call it a piecewise function.
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And according to our table, weβre looking to find the value of π of π₯ when π₯ is negative three.
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So thatβs π of negative three.
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We also want to find π of zero and π of three.
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And so we need to pay extra careful attention to the part of the function weβre going to use for each value of π₯.
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Letβs begin with π of negative three.
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Here, π₯ is equal to negative three.
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And so, since negative three is less than negative two, we need to use the first part of our function, that is, two to the power of π₯.
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And so to find π of negative three, weβre going to substitute π₯ equals negative three into that part of the function.
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And we get π of negative three is two to the power of negative three.
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And at this stage, we might recall that a negative power tells us to find the reciprocal.
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So π to the power of negative π, for instance, is one over π to the power of π.
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And this means then that two to the power of negative three is one over two cubed, which is equal to one over eight.
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And so the first value in our table is one-eighth.
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Letβs repeat this process for π₯ equals zero.
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This time zero is between negative two and three, so weβre going to use the second part of our function.
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And so, π of zero is found by substituting π₯ equals zero into the function π of π₯ equals three to the power of π₯.
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So thatβs three to the power of zero.
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Now, of course, at this stage, we might recall that anything to the power of zero is equal to one, so π to the power of zero is simply equal to one.
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And thatβs the second value in our table.
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Letβs repeat this process for the third and final column in our table.
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The third part of our piecewise function is used when π₯ is greater than or equal to three.
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So weβre going to use this value when π₯ is equal to three.
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And this means that π of three is two cubed, which is simply equal to eight.
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And so we pop eight in the final part of our table.
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The missing table values for our piecewise function π of π₯ are one-eighth, one, and eight.
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Weβre now going to look at a couple of examples on evaluating a function at a point given its graph.
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Determine π(0) using the graph.
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Letβs look at the graph of our function.
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It must be a piecewise function, and this is because itβs made up of pieces of graphs of various functions over various intervals on π₯.
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For instance, letβs take the first portion of graph here.
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This portion is defined by a specific function over the interval from negative 10 to eight.
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In fact, we could even define this as the left-closed, right-open interval.
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And thatβs because the solid dot tells us that itβs defined at π₯ equals negative 10 but not defined by this part of the function at π₯ equals negative eight.
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Then, the second part of the function allows us to define π of π₯ at π₯ equals negative eight with the solid dot here.
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But when π₯ is equal to zero, we canβt use this part of the graph to determine π of zero.
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The empty dot tells us itβs not defined by this part of our function.
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So, how are we going to determine π of zero?
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Well, when π₯ is zero, weβre looking for a part of the function that lies on the π¦-axis.
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Weβve already seen this canβt be defined by this function here, but we do have a solid dot here.
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And so, the function is actually defined when π₯ is equal to zero.
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This point has coordinates zero, four.
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So we see that π of zero must be equal to four.
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Letβs consider one further example of this type.
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Determine π of one.
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Here we see the function π of π₯ is a piecewise function.
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We know this because itβs made up of the graphs of two different functions.
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The first part of the graph is defined over the interval from negative two to one.
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But of course, this empty circle tells us that itβs not defined by this graph at this point.
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Then the second part of the function is defined over the interval from one to eight.
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Once again, the empty dot tells us that itβs not defined at π₯ equals one by this portion of the graph.
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And so, how are we going to determine π of one?
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Well, we canβt.
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π of one is totally undefined according to our graph.
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There are plenty of values that we can find.
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For example, π of five is equal to one, as is π of negative one.
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But π of one is completely undefined according to our graph.
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In this video, we learned that a piecewise function is a function made up from pieces of more than one different function.
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We saw that each function is defined on some given interval.
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Finally, we saw that we can evaluate a piecewise function using the graph or its individual function parts, but we must be careful to ensure the function is actually defined at that point.