Piecewise Function Worksheet with Answers

Understanding Piecewise Functions

Piecewise functions are a type of mathematical function that is defined by multiple sub-functions, each applied to a specific interval of the domain. These functions are used to model real-world situations where the behavior of the system changes depending on the input value.

What is a Piecewise Function?

A piecewise function is a function that is defined by multiple sub-functions, each applied to a specific interval of the domain. The general form of a piecewise function is:

f(x) = { g(x) if x is in the interval [a, b) h(x) if x is in the interval [b, c) … }

where g(x), h(x),… are sub-functions that define the behavior of the function on each interval.

Example of a Piecewise Function

Consider the following piecewise function:

f(x) = { x^2 if x < 0 x + 1 if 0 ≤ x < 2 3x - 2 if x ≥ 2 }

This function is defined by three sub-functions:

• x^2 for x < 0
• x + 1 for 0 ≤ x < 2
• 3x - 2 for x ≥ 2

Evaluating Piecewise Functions

To evaluate a piecewise function, we need to determine which sub-function to apply based on the input value. For example, to evaluate f(1), we would apply the sub-function x + 1, since 1 is in the interval [0, 2).

1. Identify the interval that the input value belongs to.
2. Apply the corresponding sub-function to the input value.

📝 Note: Make sure to identify the correct interval and apply the correct sub-function when evaluating a piecewise function.

Graphing Piecewise Functions

To graph a piecewise function, we need to graph each sub-function on its corresponding interval. The resulting graph will be a combination of the graphs of the individual sub-functions.

Consider the piecewise function:

f(x) = { x^2 if x < 0 x + 1 if 0 ≤ x < 2 3x - 2 if x ≥ 2 }

To graph this function, we would graph the following:

• x^2 for x < 0
• x + 1 for 0 ≤ x < 2
• 3x - 2 for x ≥ 2

The resulting graph would be a combination of the graphs of these three sub-functions.

Piecewise Function Worksheet

Now it’s your turn to practice! Try to evaluate and graph the following piecewise functions:

1. f(x) = { x + 2 if x < -1 2x - 1 if -1 ≤ x < 1 x^2 - 1 if x ≥ 1 }

a. Evaluate f(-2)

b. Evaluate f(0)

c. Evaluate f(2)

d. Graph the function f(x)

1. f(x) = { 2x + 1 if x < 0 x - 2 if 0 ≤ x < 3 x^2 + 1 if x ≥ 3 }

a. Evaluate f(-1)

b. Evaluate f(2)

c. Evaluate f(4)

d. Graph the function f(x)

1. f(x) = { x^2 - 1 if x < 1 2x - 2 if 1 ≤ x < 3 x + 1 if x ≥ 3 }

a. Evaluate f(0)

b. Evaluate f(2)

c. Evaluate f(4)

d. Graph the function f(x)

Answers

1. f(x) = { x + 2 if x < -1 2x - 1 if -1 ≤ x < 1 x^2 - 1 if x ≥ 1 }

a. f(-2) = (-2) + 2 = 0

b. f(0) = 2(0) - 1 = -1

c. f(2) = (2)^2 - 1 = 3

d. Graph:

|x| |f(x)|

1. f(x) = { 2x + 1 if x < 0 x - 2 if 0 ≤ x < 3 x^2 + 1 if x ≥ 3 }

a. f(-1) = 2(-1) + 1 = -1

b. f(2) = 2 - 2 = 0

c. f(4) = (4)^2 + 1 = 17

d. Graph:

|x| |f(x)|

1. f(x) = { x^2 - 1 if x < 1 2x - 2 if 1 ≤ x < 3 x + 1 if x ≥ 3 }

a. f(0) = (0)^2 - 1 = -1

b. f(2) = 2(2) - 2 = 2

c. f(4) = 4 + 1 = 5

d. Graph:

|x| |f(x)|