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Piecewise Defined Functions
Most of the functions that we’ve looked at this semester can be expressed as a single equation. For example, f (x) = 3x 2 5 x + 2, or g(x) =
p x 1, or h(x) = e 3 x^ 1. Sometimes an equation can’t be described by a single equation, and instead we have to describe it using a combination of equations. Such functions are called piecewise defined functions, and probably the easiest way to describe them is to look at a couple of examples.
First example. The function g : R! R is defined by
g(x) =
x 2 1 if x 2 ( 1, 0]; x 1 if x 2 [0, 4]; 3 if x 2 [4, 1 ). The function g is a piecewise defined function. It is defined using three functions that we’re more comfortable with: x 2 1, x 1, and the constant function 3. Each of these three functions is paired with an interval that appears on the right side of the same line as the function: ( 1, 0], and [0, 4], and [4, 1 ) respectively. If you want to find g(x) for a specific number x, first locate which of the three intervals that particular number x is in. Once you’ve decided on the correct interval, use the function that interval is paired with to determine g(x). If you want to find g(2), first check that 2 2 [0, 4]. Therefore, we should use the equation g(x) = x 1, because x 1 is the function that the interval [0, 4] is paired with. That means that g(2) = 2 1 = 1. To find g(5), notice that 5 2 [4, 1 ). That means we should be looking at the third interval used in the definition of g(x), and the function paired with that interval is the constant function 3. Therefore, g(5) = 3. Let’s look at one more number. Let’s find g(0). First we have to decide which of the three intervals used in the definition of g(x) contains the number
- Notice that there’s some ambiguity here because 0 is contained in both the interval ( 1, 0] and in the interval [0, 4]. Whenever there’s ambiguity, choose either of the intervals that are options. Either of the functions that these intervals are paired with will give you the same result. That is, 0^2 1 = 1 is the same number as 0 1 = 1, so g(0) = 1.
To graph g(x), graph each of the pieces of g. That is, graph g : ( 1, 0]! R where g(x) = x 2 1, and graph g : [0, 4]! R where g(x) = x 1, and graph g : [4, 1 )! R where g(x) = 3. Together, these three pieces make up the graph of g(x).
Graph of g : ( 1, 0]! R where g(x) = x 2 1.
Graph of g : [0, 4]! R where g(x) = x 1.
Graph of g : [4, 1 )! R where g(x) = 3.
To graph g(x), graph each of the pieces of g. That is, graph g : ( ⇤, 0] ⇥ R where g(x) = x^2 1, and graph g : [0, 4] ⇥ R where g(x) = x 1, and graph g : [4, ⇤) ⇥ R where g(x) = 3. Together, these three pieces make up the graph of g(x).
Graph of g : ( ⇤, 0] ⇥ R where g(x) = x^2 1.
Graph of g : [0, 4] ⇥ R where g(x) = x 1.
Graph of g : [4, ⇤) ⇥ R where g(x) = 3.
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To graph g(x), graph each of the pieces of g. That is, graph g : ( ⇤, 0] ⇥ R where g(x) = x^2 1, and graph g : [0, 4] ⇥ R where g(x) = x 1, and graph g : [4, ⇤) ⇥ R where g(x) = 3. Together, these three pieces make up the graph of g(x).
Graph of g : ( ⇤, 0] ⇥ R where g(x) = x^2 1.
Graph of g : [0, 4] ⇥ R where g(x) = x 1.
Graph of g : [4, ⇤) ⇥ R where g(x) = 3.
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To graph g(x), graph each of the pieces of g. That is, graph g : ( ⇤, 0] ⇥ R where g(x) = x^2 1, and graph g : [0, 4] ⇥ R where g(x) = x 1, and graph g : [4, ⇤) ⇥ R where g(x) = 3. Together, these three pieces make up the graph of g(x).
Graph of g : ( ⇤, 0] ⇥ R where g(x) = x^2 1.
Graph of g : [0, 4] ⇥ R where g(x) = x 1.
Graph of g : [4, ⇤) ⇥ R where g(x) = 3.
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Graph of the second piece of f (x): a single giant dot whose x-coordinate equals 3.
Graph of both pieces, and hence the entire graph, of f (x).
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Graph of the second piece of f (x): a single giant dot.
Graph of both pieces, and hence the entire graph, of f (x).
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Graph of both pieces, and hence the entire graph, of f (x).
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Absolute value
The most important piecewise defined function in calculus is the absolute value function that is defined by
|x| =
x if x 2 ( 1, 0]; x if x 2 [0, 1 ). The domain of the absolute value function is R. The range of the absolute value function is the set of non-negative numbers. The number |x| is called the absolute value of x. For examples of how this function works, notice that | 4 | = 4, | 0 | = 0, and | 3 | = ( 3) = 3. If x is positive or 0, then the absolute value of x is x itself. If x is negative, then |x| is the positive number that you’d get from “erasing” the negative sign: | 10 | = 10 and | 12 | = 12.
Graph of the absolute value function.
Another interpretation of the absolute value function, and the one that’s most important for calculus, is that the absolute value of a number is the same as its distance from 0. That is, the distance between 0 and 5 is | 5 | = 5, the distance between 0 and 7 is | 7 | = 7, and the distance between 0 and 0 is | 0 | = 0.
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Absolute value The most important piecewise defined function in calculus is the absolute value function that is defined by
|x| =
x if x ⇤ ( ⇥, 0]; x if x ⇤ [0, ⇥). The domain of the absolute value function is R. The range of the absolute value function is the set of non-negative numbers. The number |x| is called the absolute value of x. For examples of how this function works, notice that | 4 | = 4, | 0 | = 0, and | 3 | = 3. If x is positive or 0, then the absolute value of x is x itself. If x is negative, then |x| is the positive number that you’d get from “erasing” the negative sign: | 10 | = 10 and | 12 | = 12.
Graph of the absolute value function.
Another interpretation of the absolute value function, and the one that’s most important for calculus, is that the absolute value of a number is the same as its distance from 0. That is, the distance between 0 and 5 is | 5 | = 5, the distance between 0 and 7 is | 7 | = 7, and the distance between 0 and 0 is | 0 | = 0.
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Absolute value The most important piecewise defined function in calculus is the absolute value function that is defined by
|x| =
x if x ⇤ ( ⇥, 0]; x if x ⇤ [0, ⇥). The domain of the absolute value function is R. The range of the absolute value function is the set of non-negative numbers. The number |x| is called the absolute value of x. For examples of how this function works, notice that | 4 | = 4, | 0 | = 0, and | 3 | = 3. If x is positive or 0, then the absolute value of x is x itself. If x is negative, then |x| is the positive number that you’d get from “erasing” the negative sign: | 10 | = 10 and | 12 | = 12.
Graph of the absolute value function.
Another interpretation of the absolute value function, and the one that’s most important for calculus, is that the absolute value of a number is the same as its distance from 0. That is, the distance between 0 and 5 is | 5 | = 5, the distance between 0 and 7 is | 7 | = 7, and the distance between 0 and 0 is | 0 | = 0.
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Absolute value The most important piecewise defined function in calculus is the absolute value function that is defined by
|x| =
x if x ⇥ 0; x if 0 ⇥ x. The domain of the absolute value function is R. The range of the absolute value function is the set of non-negative numbers. The number |x| is called the absolute value of x. For examples of how this function works, notice that | 4 | = 4, | 0 | = 0, and | 3 | = 3. If x is positive or 0, then the absolute value of x is x itself. If x is negative, then |x| is the positive number that you’d get from “erasing” the negative sign: | 10 | = 10 and | 12 | = 12.
Graph of the absolute value function.
Another interpretation of the absolute value function, and the one that’s most important for calculus, is that the absolute value of a number is the same as its distance from 0. That is, the distance between 0 and 5 is | 5 | = 5, the distance between 0 and 7 is | 7 | = 7, and the distance between 0 and 0 is | 0 | = 0.
Notice in the above paragraph that the precise number 5 wasn’t really important for the problem. We could have replaced 5 with any positive number c to obtain the following translation.
|x| < c means c < x < c
For example, writing |x| < 2 means the same thing as writing 2 < x < 2, and | 2 x 3 | < 13 means the same as 13 < 2 x 3 < 13. We can use the above rule to help us solve some inequalities that involve absolute values.
Problem. Solve for x if | 3 x + 4| < 2.
Solution. We know from the explanation above that 2 < 3 x + 4 < 2.
Subtracting 4 from all three of the quantities in the previous inequality yields 2 4 < 3 x < 2 4, and that can be simplified as 6 < 3 x < 2. Next divide by 3, keeping in mind that dividing an inequality by a neg- ative number “flips” the inequalities. The result will be ^63 > x > ^23 , which can be simplified as 2 > x > 23. That’s the answer.
The inequality 2 > x > 23 could also be written as 23 < x < 2, or as x 2 ( 23 , 2).
Problem. Solve for x if | 2 x 1 | < 3.
Solution. Write the inequality from the problem as 3 < 2 x 1 < 3.
Add 1 to get 2 < 2 x < 4, and divide by 2 to get 1 < x < 2.
If c is a positive number, |x| > c means that the distance between x and 0 is greater than c. There are two ways that the distance between x and 0 can be greater than c. Either x < c or x > c.
|x| > c means either x < c or x > c
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Exercises
1.) Suppose f (x) is the piecewise defined function given by
f (x) =
x + 1 if x 2 ( 1, 2); x + 3 if x 2 [2, 1 ).
What is f (0)? What is f (10)? What is f (2)?
2.) Suppose g(x) is the piecewise defined function given by
g(x) =
3 if x 2 [1, 5]; 1 if x 2 (5, 1 ).
What is g(1)? What is g(100)? What is g(5)?
3.) Suppose h(x) is the piecewise defined function given by
h(x) =
5 if x 2 (1, 3]; x + 2 if x 2 [3, 8).
What is h(2)? What is h(7)? What is h(3)?
4.) Suppose f (x) is the piecewise defined function given by
f (x) =
2 if x 2 [ 3 , 0); e x^ if x 2 [0, 2]; 3 x 2 if x 2 (2, 1 ).
What is f ( 2)? What is f (0)? What is f (2)? What is f (15)?
5.) Suppose g(x) is the piecewise defined function given by
g(x) =
(x 1) 2 if x 2 ( 1, 1]; log (^) e (x) if x 2 [1, 5]; log (^) e (5) if x 2 [5, 1 ).
What is g(0)? What is g(1)? What is g(5)? What is g(20)?
6.) Suppose h(x) is the piecewise defined function given by
h(x) =
e x^ if x 6 = 2; 1 if x = 2.
What is h(0)? What is h(2)? What is h
log (^) e (17)
7.) Write the following numbers as integers: | 8 5 |, | 10 5 |, and | 5 5 |. The function |x 5 | measures the distance between x and which number?
8.) Write the following numbers as integers: | 1 2 |, | 3 2 |, and | 2 2 |. The function |x 2 | measures the distance between x and which number?
9.) Write the following numbers as integers: |3 + 4|, | 1 + 4|, | 4 + 4|. The function |x + 4| measures the distance between x and which number?
10.) The function |x y| measures the distance between x and which number?
11.) Solve for x if | 5 x 2 | < 7.
12.) Solve for x if | 3 x + 4| < 1.
13.) Solve for x if | 2 x + 3| < 5.
14.) Solve for x if |x + 3| > 2.
15.) Solve for x if | 4 x| > 12.
16.) Solve for x if | 2 x + 4| > 8.
Match the functions with their graphs.
21.) f (x) = e x^ 23.) p(x) =
e x^ if x 6 = 1; 2 if x = 1.
22.) g(x) = 2 24.) q(x) =
e x^ if x 6 = 1; 2 if x = 1.
A.) B.)
C.) D.)
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Match the functions with their graphs.
25.) f (x) =
p x + 2 27.) p(x) =
2 if x 2 ( 1, 0); p x + 2 if x 2 [0, 1 ).
26.) g(x) = 2 28.) q(x) =
2 if x 2 ( 1, 1); p x + 2 if x 2 [1, 1 ).
A.) B.)
C.) D.)
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