Operations with Polynomials

Transcription

1 GSE Advanced Algebra Operations with Polynomials Operations with Polynomials Operations and Composition of Functions Operations can be done with functions just like with expressions. Here is the notation we will use. The examples will use the functions f(x)= 2x + 3 and g(x) = x² x. Operation Notation Example Addition Subtraction Multiplication Division simplified, cannot be Also, in division remember that denominators cannot be zero, so g(x) 0. These "combinations" of functions use the same properties you learned in previous modules. You can find the domain of each by determining what values can be used for x, the same as in previous modules. Always simplify your answers, where possible. For more examples, see Add & Subtract, Multiply, and Divide in the sidebar. Self Check

2 For 1 4 below, use the following information to match the operation with the resulting function and domain. Match the operation with the resulting function and domain given the information above: For 5 8 below, use the following: Match the operation with the resulting function and domain given the information above: Composite Functions The next thing to look at is composite functions. Composition of functions is when we are given two functions, their composite (combined function) uses the output from one function as the input for the other function. We will see the common notation of f(x) and g(x), representing the two different functions. Notation for composite functions is The second notation allows us to see what the input is and what the output is. In f(g(x)), the output for the function g(x) is used as the input for f(x). In g(f(x)), the output for the function f(x) is used as the input for g(x). The domain of composite functions is determined by the domains of the original functions, not the resulting function. The purpose of composing functions is often to evaluate the result for a specific number. For example, if f(x) = 2x + 3 and g(x) = x² x, we might want to find. To do this, we don't have to find f(g(x)).we can simply follow order of operation, doing what is in the ( ) first. So we find g(3) by putting 3 in for x in the g(x) function. g(3) = 3² 3 = 9 3 = 6 Now we take g(3) which is 6 and put it in for x in the f(x) function. f(g(3)) = f(6) = 2(6) + 3 = 15 For more on composition of functions and for practice, see Composite and Evaluate Composite in the sidebar.

3 Let's see how this is used. Aisha made a chart of the experimental data for her science project and showed it to her science teacher. The teacher was complimentary of Aisha's work but suggested that, for a science project, it would be better to list the temperature data in degrees Celsius rather than degrees Fahrenheit. a. Aisha found the formula for converting from degrees Fahrenheit to degrees Celsius. Use this formula to convert freezing (32 F) and boiling (212 F) to degrees Celsius. SOLUTION b.later Aisha found a scientific journal article related to her project and planned to use information from the article on her poster for the school science fair. The article included temperature data in degrees Kelvin. Aisha talked to her science teacher again, and they concluded that she should convert her temperature data again this time to degrees Kelvin. The formula for converting degrees Celsius to degrees Kelvin is K = C Use this formula and the results of part a to express freezing and boiling in degrees Kelvin. SOLUTION c. Use the formulas from part a and part b to convert the following to K 238 F, 5000 F. SOLUTION In converting from degrees Fahrenheit to degrees Kelvin, you used two functions, the function for converting from degrees Fahrenheit to degrees Celsius and the function for converting from degrees Celsius to degrees Kelvin, and a procedure that is the key idea in the composition of functions. We now explore how the temperature conversions from Item 1, part c, provide an example of a composite function. d. The definition of composition of functions indicates that we start with a value, x, and first use this value as input to the function g. In our temperature conversion, we started with a temperature in degrees Fahrenheit and used the formula to convert to degrees Celsius, so the function g should convert from Fahrenheit to Celsius. What is the meaning of x and what is the meaning of g(x) when we use this notation? SOLUTION

4 e. In converting temperature from degrees Fahrenheit to degrees Kelvin, the second step is converting a Celsius temperature to a Kelvin temperature. The function f should give us this conversion; thus, f(x) = x What is the meaning of x and what is the meaning of f (x) when we use this notation? SOLUTION f. Calculate What is the meaning of this number? SOLUTION g. Calculate, and simplify the result. What is the meaning of x and what is the meaning of? SOLUTION h. Calculate using the formula from part d. Does your answer agree with your calculation from part c? SOLUTION i. Calculate, and simplify the result. What is the meaning of x? What meaning, if any, relative to temperature conversion can be associated with the value of? SOLUTION Inverse Functions In this topic, we will be looking at the inverse of a function. In order to do that, we must review what makes a function and the concept of one to one. Remember the definition of a function for every one number in the domain, there is one unique number in the range. In other words, each x can have only one y associated with it. You have used the vertical line test to determine if a graph is a function. The test states that if every vertical line intersects the graph at no more than one point, then it is a function. What is a one to one function? This is a function in which each y also has only one x associated with it. In other words, one x relates only to one y and one y relates only to one x. The test to determine one to one is the horizontal line test. The horizontal line test requires that every horizontal line intersect the graph at no more than one point. Only functions that are one to one have an inverse. For the inverse of a function, the domain and range values switch creating a new "function." If f(x) has the points (1, 4), (2, 5) and (3, 6),

5 then the inverse function, denoted will have the points (4, 1), (5, 2), and (6, 3). With a function in equation form, find the inverse by switching x and y and then solving for y. Example To verify that functions are inverses, show that Example Using the function and inverse above

6 Note "Verify" is essentially a proof, so you must include each step as you simplify. Graphs of Inverse Functions The graphs of inverse functions are symmetric across the line y = x. Using the function and inverse above, the graph is The video showcase will walk you through inverses. Cryptography Inverse functions are used by government agencies and other businesses to encode and decode information. These functions are usually very complicated. A simplified example involves the function. If each letter of the alphabet is assigned a numerical value according to its position (A = 1, B = 2,..., Z = 26), the word ALGEBRA would be encoded by putting the numbers for each letter into the function, getting The "message" can be decoded by finding the inverse function and plugging the encoded numbers in to find the numbers corresponding to the letters. a. What is the inverse of this function? SOLUTION b. What numbers do you get when you put the encoded number into the inverse? SOLUTION

7 c. What are the letters that match these numbers? SOLUTION For more on inverses, see Find Inverses and Inverses in the sidebar. Also in the sidebar, Inv and Comp will review inverses and composition of functions. Boundless is a site that covers several of the topics we have been studying in this module. Scroll down to find a topic you want to review. Compositions and Inverses of Functions Quiz It is now time to complete the "Compositions and Inverses of Functions" quiz. You will have a limited amount of time to complete your quiz; please plan accordingly. Quadratics & Operations with Polynomials Test It is now time to complete the "Quadratics & Operations with Polynomials" test. You will have a limited amount of time to complete your quiz; please plan accordingly.