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Evaluation and composition of function Evaluation: • The letter f in the function notation f(x) as the name of the function. Instead of using the equation to describe the function, • we can write . Here, f is the name of the function and f(x) is the value of the function at x. So is the value of the function at 2. • The notation affords a convenient way of prompting the evaluation of a function for a particular value of x. • Any letter can be used as the independent variable in a function. So the above function could be written . • The independent variable in a function is just a place holder. The function could be written without a variable as follows • The function can be viewed as an input/output operation. • In addition to plugging numbers into functions, we can plug expressions into functions. Plugging y + 1 into the function yields • Plug other expressions in terms of x into a function. Plugging 2x into the function yield • The variable x in the function is being replaced by the same variable. But the x in function is just a placeholder. If the placeholder were removed from the function, the substitution would appear more natural. In , we plug 2x into the left side f(2x) and it returns the right side Composition • We have plugged numbers into functions and expressions into functions; now let’s plug in other functions. Since a function is identified with its expression. • In the example above with and 2x, let’s call 2x by the name g(x). In other words, g(x) = 2x. Then the composition of f with g (that is plugging g into f) is • The notation f(g(x)) on the test. But you probably will see one or more problems that ask you perform the substitution. For another example, let and let . Then and • The composition of functions merely substitutes one function into another, these problems can become routine. Notice that the composition operation f(g(x)) is performed from the inner parentheses out, not from left to right. In the operation f(g(2)), the number 2 is first plugged into the function g and then that result is plugged in the function f. • A function can also be composed with itself. That is, substituted into itself. Let . Then Example:1 Given f(x) = 2x + 3 and g(x) = –x2 + 5, find ( f o g)(x). A) –2x2 + 13 B) 2x2 + 13 C) –2x2 – 13 D) 2x2 – 13 Solution: plugging the formula for g(x) into the formula for f(x) ( f o g)(x) = f (g(x)) = f (–x2 + 5) = 2( )+3 ... setting up to insert the input formula = 2(–x2 + 5) + 3 = –2x2 + 10 + 3 = –2x2 + 13 Example:2 Given f(x) = 2x + 3 and g(x) = –x2 + 5, find ( f o g)(1) (A) 11 (B) -11 (C) 12 (D) -12 (E) 13 Solution: ( f o g)(1) = f (g(1)) = f (–( )2 + 5) ... setting up to insert the original input = f (–(1)2 + 5) = f (–1 + 5) = f (4) = 2( ) + 3 ... setting up to insert the new input = 2(4) + 3 =8+3 = 11 Practice Questions 1) The graph of y = f(x) is shown to the right. If f(-1) = v, then which one of the following could be the value of f(v) ? (A) 0 (B) 1 (C) 2 (D) 2.5 (E) 3 2) In the function above, if f(k) = 2, then which one of the following could be a value of k ? (A) -1 (B) 0 (C) 0.5 (D) 2.5 (E) 4 3) Given f(x) = 2x + 3 and g(x) = –x2 + 5, find (g o f )(x). (A) 4x2 + 12x – 4 (B) –4x2 – 12x – 4 (C) –4x2 – 12x + 4 (D) 4x2 + 12x + 4 (E) –4x2 + 12x + 4 4) Given f(x) = 2x + 3 and g(x) = –x2 + 5, find ( f o f )(x) (A) 4x – 9 (B) 4x + 7 (C) 4x + 9 (D) -4x + 9 (E) -4x - 9 5) Given f(x) = 2x + 3 and g(x) = –x2 + 5, find (g o f )(1) (A) 20 (B) 18 (C) 19 (D) -20 (E) 21 NEED HELP? Visit www.educateNcare.com Found an error in the worksheet? Email us at info@educateNcare.com and get an awesome FREEBIE!!

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posted: | 11/15/2010 |

language: | English |

pages: | 5 |

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Free SAT worksheet on evaluation and composition of functions. The worksheet includes explanation, solved example problems and practice problems

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