Introduction A piecewise function is a function that is defined by two or more expressions on separate portions of the domain. Piecewise functions are.

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Introduction A piecewise function is a function that is defined by two or more expressions on separate portions of the domain. Piecewise functions are useful for summarizing a real- world scenario that has multiple conditions. For instance, the amount withheld from an employee’s paycheck to fund Social Security benefits depends on that individual’s income. If a person’s annual income is less than or equal to a certain limit (which changes each year), then that person pays a percentage of that annual income toward Social Security. People earning more than that limit pay a flat rate or specific amount instead of a percentage. 6.2.1: Piecewise Functions

Introduction, continued In a graph of this situation, the “pieces” of the graph representing the different tax rates would differ in appearance. In this lesson, you will learn how to create and graph piecewise functions to solve everyday problems, such as determining how much in Social Security taxes you will pay based on your salary. 6.2.1: Piecewise Functions

Key Concepts A piecewise function can be continuous, with no break in the graph of the function across a specified domain, or discontinuous, in which the graph has a break, hole, or jump. 6.2.1: Piecewise Functions

Key Concepts, continued The notation used to represent the pieces of the same function over different restricted domains is a brace: {. When stating the domain of the entire piecewise function, write the domain for each individual piece next to the appropriate expression. For example, for the piecewise function f(x): 6.2.1: Piecewise Functions

Key Concepts, continued One piece of the function, f(x) = 4.57x, represents the portion of the function over the restricted domain x ≤ 10. The other piece, f(x) = 3.43x, represents the portion of the function for which x > 10. On a graph, points of discontinuity or “holes” are shown as open circles, whereas closed circles are used to denote defined endpoints. 6.2.1: Piecewise Functions

Common Errors/Misconceptions misinterpreting variables from a word problem defining incorrect intervals for the functions incorrectly evaluating the function pieces at specified intervals incorrectly noting a closed or open circle on a graph 6.2.1: Piecewise Functions

Guided Practice Example 1 Suzanne has a part-time job writing blog posts, which pays $70 per post if she submits fewer than 5 posts a week. If she posts more than 5 blogs in a week, Suzanne is paid 1.5 times the regular amount for each additional post. Create a piecewise function to represent Suzanne’s income using this pay scale. How much will Suzanne be paid if she submits 4 posts in the next week? How much will she be paid if she submits 6 posts? 6.2.1: Piecewise Functions

Guided Practice: Example 1, continued Write the equations that make up the piecewise function. Use the values in the original problem to create the equations. Suzanne’s income is $70 per post, or 70x when the number of posts submitted is less than or equal to 5. Therefore, the first equation is f(x) = 70x. 6.2.1: Piecewise Functions

Guided Practice: Example 1, continued When Suzanne submits more than 5 posts, her income is $70 per post for the first 5 posts plus 1.5 times the regular pay for each additional post. Symbolically, this can be written as 70 • 5 + 1.5[70(x – 5)]. Therefore, the second equation is f(x) = 350 + 105x – 525, which simplifies to f(x) = 105x – 175. 6.2.1: Piecewise Functions

Guided Practice: Example 1, continued Determine the domain for each piece of the function. Start with the domain for which the first piece, f(x) = 70x, is valid. The pay rate of $70 is only valid when x ≤ 5. Since Suzanne has to write at least 1 post to be paid, the domain is also limited to values greater than or equal to 1. Therefore, the domain of f(x) = 70x is 1 ≤ x ≤ 5. Find the domain for the second piece, f(x) = 105x – 175. The second piece is valid when Suzanne posts more than 5 blogs. Therefore, the domain of f(x) = 105x – 175 is x > 5. 6.2.1: Piecewise Functions

Guided Practice: Example 1, continued Combine the results of steps 1 and 2 to write the piecewise function. When writing a piecewise function, write the equation of each piece to the right of a brace, with its domain next to it. Each piece and domain is grouped on a separate line. The piecewise function that represents Suzanne’s weekly income is 6.2.1: Piecewise Functions

Guided Practice: Example 1, continued Note that x is not less than 1, since she cannot submit less than a whole post, nor can she submit a negative post. Therefore, in either piece of the function, x is greater than or equal to 1. 6.2.1: Piecewise Functions

Guided Practice: Example 1, continued Use the piecewise function to calculate Suzanne’s income for 4 posts in one week. If 4 posts are submitted, Suzanne’s income is based on the first equation of the piecewise function, f(x) = 70x, because 4 ≤ 5. Substitute 4 for x in the first piece of the function and then solve. f(x) = 70x First piece of the function f(4) = 70(4) Substitute 4 for x. f(4) = 280 Simplify. Suzanne’s income for submitting 4 posts in one week is $280. 6.2.1: Piecewise Functions

Guided Practice: Example 1, continued Use the piecewise function to calculate Suzanne’s income for 6 posts. For 6 posts, Suzanne’s income is based on the second equation of the piecewise function, f(x) = 105x – 175, because 6 > 5. Substitute 6 for x in the second piece of the function and then solve. 6.2.1: Piecewise Functions

✔ Guided Practice: Example 1, continued f(x) = 105x – 175 Second piece of the function f(6) = 105(6) – 175 Substitute 6 for x. f(6) = 630 – 175 Simplify. f(6) = 455 Subtract. Suzanne’s income for submitting 6 posts in one week is $455. ✔ 6.2.1: Piecewise Functions

Guided Practice: Example 1, continued http://www.walch.com/ei/00743 6.2.1: Piecewise Functions

Guided Practice Example 3 Use the following graph to identify the domain of the function and where the graph is continuous and discontinuous. 6.2.1: Piecewise Functions

Guided Practice: Example 3, continued Identify the domain of the piecewise function. The domain of the function is the x-values for which the function exists. It can be seen from the graph that the function exists on the intervals (0, 1] and (2, ∞). 6.2.1: Piecewise Functions

✔ Guided Practice: Example 3, continued Use the graph to identify where the function is continuous and discontinuous. The function is continuous on intervals where the graph is a single unbroken curve. Therefore, the function is continuous on the intervals (0, 1], (2, 5], and (5, ∞). The function is discontinuous at the x-values that are represented by a break or a jump in the graph. Therefore, this function is discontinuous on the intervals (–∞, 0), (1, 2], and at the point x = 5. ✔ 6.2.1: Piecewise Functions

Guided Practice: Example 3, continued http://www.walch.com/ei/00744 6.2.1: Piecewise Functions