Lesson 6-2: Function Operations

Transcription

1 So numbers not only have a life but they have relationships well actually relations. There are special relations we call functions. Functions are relations for which each input has one and only one output. Every time you plug something in, you get exactly the same thing out. Functions are predictable. Okay, that s all well and good, but what can you do with a function? Valid function operations You can do all the normal math stuff with functions. You can add, subtract, multiply and divide them getting a new function. For explanation purposes, lets use the following two functions: f(x) = x + 1 and g(x) = x 3. Again, when we combine two functions, we get a new function; let s call the new function h(x). Basically all we re doing is substitution here if we see f(x) we replace it with x + 1: 1. Multiply a function by a number: h( x) 3 f ( x) Multiply function f by 3 h( x) 3(x 1) Substituteintheexp ression f ( x) h( x) 6x 3 Simplify... distributive property. Add functions: h( x) f ( x) g( x) Thetwo functions added together... h( x) (x 1) ( x 3) Just substitute... h( x) 1 x 3... and simplify h( x) 3x... by combining liketerms 3. Subtract functions: h( x) f ( x) g( x) One function subtracted from another... h( x) (x 1) ( x 3) Just substitute and simplify... h( x) x 1 x 3... be careful removing the parentheses!... h( x) x 4... by combining liketerms 4. Multiply functions: h( x) f ( x) g( x) Thetwo functions multiplied together... h( x) ( x 1)( x 3)... Just substitute and simplify... FOIL this out... h x x x x Nowcombineliketerms ( ) h x x x done ( ) ! 5. Divide functions: f( x) h( x) The1st function divided by the nd... gx ( ) x 1 h( x)... Just substitute and simplify if possible... nothing to do here... x 3 Page 1 of 5

2 Finding the domain of a function If you recall, when we have an equation for a relation or function, that equation basically tells us all of the ordered pairs for that function. For the vast majority of functions, that is an infinite number of ordered pairs. I don t know about you, but I have no interesting spending the rest of my life listing the ordered pairs for a particular function! So if I were asked to say what the domain of a particular function is, how could I answer it simply? Let s try a simple example. What is the domain of f ( x) x 1? Well, the domain is all of the possible x values the function can handle. Can you think of any x values that can t be used with this function? Nope, neither can I. So, what is the domain? The domain is all real numbers. So it sounds like to determine the domain of a function, the question I need to ask myself is are there any x-values that can t be used with this function? Let s try it on a one that is a bit trickier: what is the domain of gx ( ) x 3? Are there any x values that can t be used with this function? Hmm, this is a fraction is there anything we can t allow with a fraction? Yuppers! We can t allow a 0 in the denominator! That would give us division by zero. What value of x would give us a 0 denominator? To find out, take the expression in the denominator, set it equal to zero and solve: x 3 0 x 3 so x can t be 3. The domain is all real numbers except x = 3. How about this one? What is the domain of h( x) 3x 6? Okay, are there any x values that can t be used with this function? This is a square root. Is there anything (in the real number system) that we can t do with a square root? Oh yeah! We can t take the square root of a negative number! That means that whatever is under the square root sign has to be greater or equal to zero! This means that 3x 6 0or x. So the domain is all real numbers x. So when determining the domain, the main things to watch out for are: 1. Dividing by zero. Square root of a negative number Page of 5

3 Finding the range of a function Once we determine the domain, we can figure out the range. Often it helps to graph the function (on your calculator) to get an idea of how the function behaves. Even so, it is important that you think about how numbers work as you look at the function. With range, we want to ask the question, are there any numbers we could never get as output. Sometimes this is easy to do just by looking at the function. For instance, consider the following function. What is its range? In other words, what are all the possible values we can get out of it? h( x) 3x 6 Hmm, this is a square root. I know that for real numbers we can never get a negative answer from a square root. So that means that the only numbers we can get out of this function are non-negative real numbers. Therefore the range of this function is h(x) 0. In general, the best thing to do for now is to look at the graph of the equation to see if there are any gaps or places the graph goes away. If you aren t provided a picture, you can use your calculator for this. As an example, let s consider the graph of the function f( x) x 5 : My bet is you ve never seen a graph like this before. Relax you re not going to have to graph this right now. All we need to do is look at it to figure out what the domain and range of this function is. The domain: looking at the graph you can see the vertical dashed line. The dashed line is showing that the graph gets really close to the x value of 5 but never actually touches it. In a sense, there is a hole or gap there. Now look at the function. What Page 3 of 5

4 value of x can we never use? It is a fraction so we can t divide by zero: x = 5 causes division by 0. Since the curves look like they go infinitely left and right, the domain of is all real numbers except x = 5. Does that make sense when you compare it with the graph? The range: again, look at the graph. The upper purple and the lower green curves seem to get closer to the x-axis as you move out right and left, but never touch it. That means the graph never really touches y = 0. In a sense there is a hole there too. Both curves do however seem to go infinitely up and infinitely down. This means the range is all real numbers except y = 0. Bottom line here? Try to make sense of the function and also look at its graph. Using both together, you should be able to determine both the domain and the range. The composition of two functions There is one final operation we can do with functions. It has a strange name: composition. It basically means you plug one function into another. Another way to think of it is we re taking the function of a function. Here is how it works. If I have two functions f(x) = x + 1 and g(x) = x 3 there are two ways I could do the composition. I can take the composition of f(x) with g(x) and I can take the composition of g(x) with f(x). The main function is listed first; the one we re going to plug into the other is listed second. First a reminder: what does f(x) = x + 1 mean? It means that what ever is between the parenthesis gets plugged in for x in the right side. So if I had f(3.15) that would mean f(3.15) = (3.15) + 1. Or if I had f(tom) that would mean f(tom) = (Tom) + 1. Again, whatever I plug into the function (put inside the parentheses) gets plugged in for x on the right side. Okay, here are the two ways I can do the composition of two functions f(x) and g(x). Remember, the main function is the first; the one to plug into the other is the second. 1. I could take the composition of f(x) with g(x) yielding a new function: f ( x) 1 and g( x) x 3 f ( x) 1 The main function is f(x) f ( g( x) ) ( g( x) ) 1 Thecomposition of f(x) with g(x)...plug g(x) into f(x) ( x -3) 1 Replaceg(x) with its expression and simplify distributive property and simplify...we're done! Page 4 of 5

5 . I could also take the composition the other way: g(x) with f(x): f ( x) 1 and g( x) x 3 gx ( ) x 3 The main function is g(x) g( f ( x) ) ( f ( x) ) 3 The composition of g(x) with f(x)...plug f(x) into g(x) ( x 1) 3 Replacef(x) with its expression and simplify and simplify......we're done! Page 5 of 5