Accelerated Math 2 Name ______

Evaluating Piecewise Functions

Evaluate the following functions at f(4):

1. 2. 3.

Evaluating piecewise functions is different because there are 2 or more functions within the piecewise function. For example, if you were asked to evaluate the piecewise function

you must decide which part of the piecewise function to plug the x values into in order to evaluate the functions. The domains of each part of the piecewise functions are what let you know which equation to plug the value into. For example, if you were asked to evaluate the piecewise function above at 0, you would use the quadratic equation. If you were asked to evaluate the function at 6, you would use the linear equation. When evaluating the function at 4, you use the quadratic equation because 4 is included in the domain of the quadratic equation, but not in the domain of the linear equation.

The piecewise function above is a discontinuous function, so the domains do not overlap. When a piecewise function is continuous, the domains will overlap because there are no breaks in the graph. One example of a continuous piecewise function is the following:

You evaluate this piecewise function the same as you would evaluate the previous function. The only difference comes when evaluating the function at an x-value that is in the domain of multiple parts of the function. For example, if you were asked to evaluate the function at f(-1), you would not know which equation to substitute x for -1 into. You can evaluate this function at f(-1) using either part of the piecewise function that includes -1 in the domain and you will get the same value.

f(-1)= -2(-1) – 3 f(-1)= (-1)

2 – 3 -1

-1

Evaluating f(-1) using both equations will give you the same value! This only works in continuous piecewise functions when the value of x that you are evaluating is included in multiple domains.

Absolute Value as a Piecewise Function

Evaluating Absolute Value Equations Graphically

You can evaluate absolute value expressions algebraically and graphically. Let’s start by solving absolute value functions graphically. Absolute value equations are symmetric. Because these functions are symmetric, there will be two answers to every equation much like there were in quadratic equations.

Ex 1. Solve the equation .

/ When solving an absolute value function graphically, look at each side of the equal sign as two separate functions:
f(x)= g(x)= 4
Graph each equation on the same graph and see where they intersect.
The points where the two graphs intersect are where they are equal. Therefore, the answer to the equation is written as the x-values of the intersection points: -1, and 7.

Ex 2. Solve the equation

/ The difference between this graph and the graph on the previous page is that the absolute value is not isolated. You must isolate the absolute value before solving this equation. Thus your two equations are
f(x)= g(x)= 6
After graphing each of these functions individually, it is easy to see that these two functions are equal at x=______

Ex 3. Solve the equation

/ Looking at these graphs, you see that they do not intersect? Why does it make sense that these two equations are never equal? ______
______
______.
An absolute value function will never be equal to a negative number, as absolute value is a measure of distance and you cannot have a negative distance.

Evaluating Absolute Value Equations Algebraically

Because absolute value can be written as a piecewise function, you are really setting two equations equal to the same number. That is why you have two solutions. You have seen this graphically, but let’s see how to solve absolute value equations algebraically.

Look at example 1 from above: The parent absolute value when written as a piecewise function is . Any absolute value function can be written as a piecewise function with the equation underneath the radical, and the negation of that equation. Therefore the equation has two pieces: . Therefore in order to solve the equation you must set each part of the piecewise function equal to 4.

x – 3= 4 -(x – 3)= 4 These solutions are the same as the solutions to

+3 +3 -x + 3 = 4 the same equation when solved graphically.

x = 7 -x = 1

x = – 1

Ex 2:

- 2 - 2

x + 4 = 6 -(x + 4) = 6

- 4 - 4 - x – 4 = 6

x = 2 + 4 + 4

- x = 10

x = – 10

It is important to note when an absolute value function is set equal to a negative number. As said before you cannot have a negative distance, and therefore you cannot have a negative absolute value. Trying to solve an absolute value function that is set equal to a negative number can be deceiving. It may seem like the problem can be solved. DO NOT TRY to solve an absolute value that is set equal to a negative number.

Solve the following absolute value equations graphically AND algebraically.

1. Graphically Algebraically

/
x / y

2. Graphically Algebraically

/
x / y

3. Graphically Algebraically

/
x / y

HOMEWORK

Evaluate the following piecewise functions:

1. f(3) 2. f(0) 3. f(5) 4. f(-3)

5. g(4) 6. g(2) 7. g(0) 8. g(1)

9. h(-3) 10. h(1) 11. h(3) 12. h(5)

Evaluate the following piecewise functions by looking at the graphs.

13. 14. 15.


f(2)= f(3)= /
f(0)= f(2)= /
f(-3)= f(0)=

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