Skill 3 Relations and Functions

Transcription

1 Skill 3 Relations and Functions 3a: Use Interval and Set Notation 3b: Determine the domain and range of a relation given a set of ordered pairs, a graph, or an equation 3c: Determine whether a relation is a function given a set of ordered pairs, a graph, or an equation 3d: Find function values from a graph, a function equation (including piecewise), or given variable arguments 3e: Determine a combination of two functions Find (f ± g)(x) and domain Find f(x)g(x) and f(x) and domain g(x) Find the composition of two functions (f g)(x) and domain 3f: Find Inverse Functions Find the inverse given a set of ordered pairs, or an equation Sketch the inverse of a function given a graph Check inverses using function composition (f g)(x) = x and (g f)(x) = x One-to-One Functions

2 Skill 3a: Use Interval and Set Notation Describe, in words, the set of numbers that are represented on the number line below: Interval Notation: Set Builder Notation: Interval notation represents the set of numbers on a number line using parentheses ( ) to represent open intervals and brackets [ ] to represent closed (included) intervals. Intervals are always written with the smallest value on the left and the largest value on the right. The ends of the number lines are represented with and. Parentheses are always placed next to and because they do not represent an actual number where the interval stops. 1. Interval Notation: Set Builder Notation: 2. Interval Notation: Set Builder Notation: 3. Interval Notation: Set Builder Notation: 4. Interval Notation: Set Builder Notation: Set-Builder Notation is another way to represent an interval. Set notation uses inequality symbols to express the interval. For the number line shown at the top of the page the interval would be expressed in set notation as {x x R, x < 1} This is read The set of all numbers x, such that x is an element of the Real numbers and x is less than 1.

3 Skill 3b: Determine the domain and range of a relation given a set of ordered pairs, a graph, or an equation Peanuts sell for $5 a pound. Determine the price for each of the following amounts: 0.5 pounds, 1 pound, 2 pounds, 3.5 pounds, 5 pounds, A data set like this forms a relation. The relation shown above could be represented by a set of ordered pairs, a graph, or an equation. Write the relation above as a set of ordered pairs: The set of all of the values of the independent variable (typically x) is the domain of the relation. The set of all of the values of the dependent variable (typically y) is the range of the relation. What is the domain of the relation written as ordered pairs above? What is the range of the relation written as ordered pairs above? Determine the domain and range of each relation below. 1. {(3, 2); (4, 8); (5, -1); (0, 2); (2, 1)} 2. {(0, 2); (2, 3); (4, 2); (2, 5)} Domain: Domain: Range: Range: Domain and Range can also be written for relations that are are represented on graphs Domain: Domain: Domain: Range: Range: Range:

4 When a relation is written as an equation the domain and range can be determined by looking at the equation. Most often expressions under even indexed radicals (must be greater than or equal to zero) and expressions under the denominator (must not be equal to zero) determine the domain of an expression. Most often even indexed radicals, even powers, and absolute values determine the range of an expression. The output of these expressions are always greater than or equal to zero. Determine the Domain and Range of each of these relations. Use a graph to check your answer. 6. y = x y = 6 x y = (3x + 2) 4 10 Domain: Domain: Domain: Range: Range: Range: 9. y = x 2 + 8x y = 6x+2 x y = 2x Domain: Domain: Domain: Range: Range: Range:

5 Skill 3c: Determine whether a relation is a function given a set of ordered pairs, a graph, or an equation A function is a special type of relation. In a function every value in the domain is matched up with exactly one value in the range. A map can be used to verify this with a set of ordered pairs. Is the relation {(8, -5); (6, 3); (4, -1); (-3, 3)} a function? Determine whether each relation below represents a function. 1. {(3, 2); (4, 8); (5, -1); (0, 2); (2, 1)} 2. {(0, 2); (2, 3); (4, 2); (2, 5)} 3. Does the relation shown in the graph represent a function? Given a graph of a relation the vertical line test can be used to determine if a relation is a function. If a vertical line crosses the graph of the relation in more than one spot, the relation is not a function. Use the vertical line test to determine if the graph shown is a relation

6 To determine if a relation that is represented by an equation is a function, try to solve the expression for y. After doing so if there is only one possible value of y for any given x, the relation is a function. Determine if the relations given below are functions 7. 2x + 3y = 9 8. x + y 2 = 4 9. x 2 + y = 4 3d: Find function values from a graph, a function equation (including piecewise), or given variable arguments Given the graph of the function f(x) shown below, determine the following: 1. f(1) 2. f( 8) 3. f(0) 4. For what value(s) of x is f(x) = 4 Find the following if g(x) = x 2 2 x g(1) 6. g(4) 7. g(0) 8. g(-1) 6 x x Find the following if f(x) = { x 4 2 x < 6 6 x 12 x > f(-3) 10. f(6) 11. f(12) 12. f(13) Find the following if g(x) = x 2 3x 13. g(4a) 14. g(2a - 3) 15. g(a + h) 16. If f(x) = 3x + 2, find f(a), f(a + h), and f(a+h) f(a) h

7 3e: Determine a combination of two functions Find (f ± g)(x) and domain If f(x) = x 2 3 and g(x) = x, 1. What is f(4) + g(4) 2. What is f(4) g(4) 3. What is f(x) + g(x) 4. What is f(x) g(x) The domain of (f ± g)(x) is the intersection of the domains of f(x) and g(x). This means the domain of (f ± g)(x) includes only those values that are in both the domain of f(x) and the domain of g(x). 5. For the example above, what is the domain of (f ± g)(x)? 6. If f(x) = 3x 12 and g(x) = 9 x, what is the domain of (f + g)(x)? 7. If (x) = 5, g(x) = 4 and h(x) = f(x) g(x), determine the simplified form of h(x) and x 2 x+1 the domain of h(x)? Find f(x)g(x) and f(x) and domain g(x) If f(x) = x 2 4 and g(x) = x 2, 1. What is f(3) g(3) 2. What is f(3) g(3) 3. What is f(x) g(x) 4. What is f(x) g(x) The domain of f(x)g(x) is found in the same manner as the domain (f + g)(x). The domain of f(x) g(x) is the same but also includes the restriction that g(x) For the example above, what is the domain of f(x)g(x)? What is the domain of f(x) g(x)? 6. If f(x) = 3x 12 and g(x) = x 2, what is the domain of f(x)g(x)? 7. If (x) = x, g(x) = 6 g(x) and h(x) =, determine the simplified form of h(x) and the x+3 x 4 f(x) domain of h(x)?

8 Find the composition of two functions, (f g)(x), and the domain A function composition(f g)(x), is created by finding f(g(x)), in other words evaluating the function f(x) for the function g(x). f(x) = x 2 + x and g(x) = 2x 1. Find (f g)(x) 2. Find (g f)(x) The domain of (f g)(x) is all values of x that are in the domain of g(x) except for those values of x that result in a g(x) that is not in the domain of f(x). 3. Let f(x) = 1 and g(x) = x + 2. x 1 A) What is the domain of f(x)? B) What is the domain of g(x)? C) For what value of x does g(x) equal the answer for A)? D) The domain of (f g)(x) is the answer for B), except for your answer for C). 4. Let f(x) = 2x 5 and g(x) = x 2. Find (g f)(x) and state the domain. 5. Let f(x) = 6 x+4 6. Given the following, if 1 and g(x) =. What is the domain of (f g)(x). x+1 x f(x) g(x) A) h(x) = f(x) + g(x), find h(8) B) h(x) = f(x) g(x), find h(3) C) h(x) = f(g(x)), find h(6) D) h(x) = g(f(x)), find h(5)

9 3f: Find Inverse Functions Find the inverse given a set of ordered pairs, or an equation. In the function f(x) = 2x + 6 when x is replaced with 5, the result is 16. This gives the ordered pair (5, 16). The inverse function of f(x), written f 1 (x), undoes the function f(x). This means that when the x in the inverse function is replaced with 16, the process that was done initially is reversed and the result is 5. This means the ordered pair on the inverse is (16, 5) Thus when given a function that is a set of ordered pairs, the inverse of the function is just the set of ordered pairs with the x and y values switched. 1. State the inverse of the function {(3, 2); (4, 8); (5, -1); (0, 2); (2, 1)} To find the inverse of a function that is given as an equation, replace the function notation with the variable y. Then switch the x and y variables and solve the equation for y. Find the inverse of each function 2. f(x) = 2x g(x) = x f(x) = x 2 + 6x g(x) = 4 x+2 Sketch the inverse of a function given a graph The function f(x) is represented on the graph. Plot the ordered pairs of the inverse of the function on the same graph. How are the graphs related?

10 Since the inverse of a function has switched values for x and y, the graph of an inverse function will always be a reflection of the graph of the function across the line y = x. Use this to sketch a graph of the inverse of each function shown below Check inverses using function composition (f g)(x) = x and (g f)(x) = x Sometimes it is difficult, or maybe even impossible, to find an inverse function algebraically. However you can check to see if g(x) is in inverse of f(x) by finding f(g(x)) and g(f(x)). If the result for each of these is x, then g(x) is an inverse of f(x). 8. f(x) = 3x + 9 and g(x) = 1 x 3. Verify that g(x) is the inverse of f(x) f(x) = x+1 x and g(x) = 1 x 1. Verify that g(x) is the inverse of f(x).

11 One-to-One Functions A relation is a function if each item in the domain has only one corresponding item in the range. A function is one-to-one if each item in the range has only one corresponding item in the domain. In other words a function is one-to-one if its inverse is also a function. 1. Use a map to verify that the relation {( 2, 7); (0, 5); (1, 4); (3, 5), (4, 1)} is one-to-one. Examine the graph and determine if it represents a one-to-one function Determine if the following equations are one-to-one 5. y = 3x 7 6. y = 2x 2 + 3x y = x 3 8. y = x + 1