1. Introduction Functions have many characteristics, such as domain, range, asymptotes, zeros, and intercepts. These functions can be compared even when given in a different format, such as when provided as an equation, graph, or table. The key to comparing functions is to either find the same information from the different forms or transfer the functions into the same form for a clear comparison. 1 3.5.1: Comparing Properties of Functions

2. Key Concepts Functions can be described numerically in a table, verbally, algebraically, or graphically. To compare two functions, determine the possible zeros, y-intercepts, domain, and range of each function. Depending on how the function is represented, this information can be found in various ways. 2 3.5.1: Comparing Properties of Functions

3. Key Concepts, continued Determining the Zeros of a Function The zero of a function is the point where a function equals 0 and at which the graph crosses the x-axis. A zero is also called the root or x-intercept of a function. To identify the zero of an equation, replace f(x) or y with 0 and solve for x. To identify the zero of a graph, look for the point where the curve crosses the x-axis. To identify the zero in a table, look for a point that has 0 as its y-coordinate. 3 3.5.1: Comparing Properties of Functions

4. Key Concepts, continued Determining the y-intercepts of a Function The y-intercept of a function is the point where the graph crosses the y-axis. It is written as (0, y). To identify the y-intercept of an equation, replace x with 0 and solve for f(x) or y. To identify the y-intercept of a graph, look for the point where the curve or line crosses the y-axis. To identify the y-intercept in a table, look for a point that has 0 as its x-coordinate. 4 3.5.1: Comparing Properties of Functions

5. Key Concepts, continued Determining the Asymptotes of a Function An asymptote is a line that a function gets closer and closer to, but never crosses or touches. In other words, it is a line that a curve approaches (but does not reach) as its x- or y-values become very large or very small. When a rational function is in equation form, find the vertical asymptote by setting the denominator equal to 0 and solving for x. 5 3.5.1: Comparing Properties of Functions

6. Key Concepts, continued To identify the vertical asymptote of a graph, look for a vertical line that the curve approaches but never reaches. To identify the vertical asymptote in a table, look for a grouping of x-coordinates with similar values. For a first-degree rational function, the horizontal asymptote can be found by dividing the coefficients of the variables. 6 3.5.1: Comparing Properties of Functions

7. Key Concepts, continued 7 3.5.1: Comparing Properties of Functions For example, in the function 3 divided by 2 is 1.5, and the asymptote is located at y = 1.5, as shown in the graph.

8. Key Concepts, continued To identify the horizontal asymptote of a graph, look for a horizontal line that the curve approaches but never reaches. To identify the horizontal asymptote in a table, look for a grouping of y-coordinates with similar values. 8 3.5.1: Comparing Properties of Functions

9. Key Concepts, continued Determining the Domain of a Function The of a function is the set of all input values (x-values) that satisfy the given function without restriction. For a radical function, the domain of an equation is found by solving an inequality in which the radicand is listed as being greater than or equal to 0. 9 3.5.1: Comparing Properties of Functions

10. Key Concepts, continued The domain of a first-degree rational function is found by identifying the asymptotes. The domain will contain every real number except the point where the vertical asymptote is defined. To identify the domain from a graph, determine the x-coordinate where the curve starts and in which direction the curve continues. Whichever values are not allowed in the domain will be vertical asymptotes on the graph. 10 3.5.1: Comparing Properties of Functions

11. Key Concepts, continued Recall that the domain of a function can be expressed in interval notation. That is, the domain is written in the form of (a, b), where a and b are the endpoints of the interval. Depending on the values of the interval, the notation may change, as shown in the following table. 11 3.5.1: Comparing Properties of Functions

12. Key Concepts, continued 12 3.5.1: Comparing Properties of Functions Interval Notation ExampleDescription (a, b)(2, 10) All numbers between 2 and 10; endpoints are not included. [a, b][2, 10] All numbers between 2 and 10; endpoints are included. (a, b](2, 10] All numbers between 2 and 10; 2 is not included, but 10 is included. [a, b)[2, 10) All numbers between 2 and 10; 2 is included, but 10 is not included.

13. Key Concepts, continued Determining the Range of a Function The range is the set of all outputs of a function. It is the set of y-values that are valid for the function. For a radical function, the best way to find the range of an equation is to first identify the domain, and then find the corresponding values for f(x) or y. The range of a first-degree rational function is found by identifying the asymptotes. The range will contain every real number except the point where the horizontal asymptote is defined. 13 3.5.1: Comparing Properties of Functions

14. Key Concepts, continued To identify the range from a graph, determine the y-coordinate where the curve starts and in which direction the curve continues. Similar to the domain, the range can also be expressed in interval notation. 14 3.5.1: Comparing Properties of Functions

15. Common Errors/Misconceptions performing incorrect processes for finding the vertical and horizontal asymptotes misidentifying the domain and range misinterpreting interval notation as coordinate notation 15 3.5.1: Comparing Properties of Functions

16. Guided Practice Example 1 Given f(x) as shown in the graph and which function has the greater zero? 16 3.5.1: Comparing Properties of Functions

17. Guided Practice: Example 1, continued 1.Identify the zero of f(x). Find the zero of a graph by identifying the point at which the curve crosses the x-axis. The curve of the graph crosses the x-axis at x = 5. Therefore, 5 is the zero of f(x). 17 3.5.1: Comparing Properties of Functions

18. Guided Practice: Example 1, continued 2.Identify the zero of g(x). To find the zero of the function g(x), set the function equal to 0 and solve for x. 18 3.5.1: Comparing Properties of Functions

19. Guided Practice: Example 1, continued 19 3.5.1: Comparing Properties of Functions The zero of g(x) is 2. Original function Substitute 0 for g(x). 0(7x + 2) = 3x – 6Cross multiply. 0 = 3x – 6Simplify. 6 = 3xAdd 6 to both sides x = 2Divide both sides by 3.

20. Guided Practice: Example 1, continued 3.Determine which function has the greater zero. The zero of f(x) is 5 and the zero of g(x) is 2. 5 is greater than 2; therefore, given the two functions f(x) and g(x), f(x) has the greater zero. 20 3.5.1: Comparing Properties of Functions ✔

21. Guided Practice: Example 1, continued 21 3.5.1: Comparing Properties of Functions

22. Guided Practice Example 4 The radical function f(x) has a domain of and a range of. The function g(x) is shown in graph. Which function has a higher y-intercept? 22 3.5.1: Comparing Properties of Functions

23. Guided Practice: Example 4, continued 1.Estimate the y-intercept of g(x). The curve of the graph appears to cross the y-axis at the point (0, 2.5). 23 3.5.1: Comparing Properties of Functions Equation step within table

24. Guided Practice: Example 4, continued 2.Estimate the y-intercept of f(x). Since the range of the function is, the function only exists at or above y = 5. Therefore, this function must intercept the y-axis somewhere above the point (0, 5). 24 3.5.1: Comparing Properties of Functions

25. Guided Practice: Example 4, continued 3.Determine which function has a higher y-intercept. Even though the exact y-intercept of f(x) cannot be determined, it is clear that it is located somewhere above the point (0, 5). Since the point (0, 2.5) is the y-intercept of g(x), and 2.5 is less than 5, it can be determined that f(x) has the higher y-intercept. 25 3.5.1: Comparing Properties of Functions ✔

26. Guided Practice: Example 4, continued 26 3.5.1: Comparing Properties of Functions