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Algebra 2 Notes Ch. 8 – Inverses & Radicals p. 1 of 6 8.1 – COMPOSITION OF FUNCTIONS If f(x) = x2 – 2x + 3, then f(5) means putting 5 in the place of x. So, f(5) = 52 – 2(5) + 3, or f(5) = ________. What is f(x+2)? _____________. How about f(3n)? ____________. Composition of Functions By f(g(x)), or the COMPOSITION of f and g, we mean the following: 1. Evaluate g(x) 2. Evaluate f of whatever the value of g(x). In other words: f(the answer to g(x)). Suppose f(x) = x + 5, and g(x) = x2. To find f(g(3)), 1. Find g(3). g(3) = 32 or 9. 2. Find f(9). f(9) = ________. So f(g(3)) = _______. ** Just as with any ( ), you work from the inside out** In order…given x, find g(x), use this to find f(g(x)) Find the following: 1. f(g(-1)) = Algebra 2 Notes Ch. 8 – Inverses & Radicals p. 2 of 6 f(x) = x + 5 g(x) = x2 2. f(g(4)) = 3. f(g(8)) = 4. g(f(8)) = 5. g(f(x)) = Alternate form: f(g(x)) can be written as (fg)(x). It still means do g(x) first. Algebra 2 Notes Ch. 8 – Inverses & Radicals p. 3 of 6 8.2 INVERSES OF RELATIONS I. What is an inverse? Very generally, an inverse “undoes” or “gets you back to where you started.” Ex. The inverse operation for addition is _______________. The inverse operation for multiplication is _______________. The inverse operation for a 3rd power is _________________. II. How do you find the inverse of a function? A. If you know the ordered pairs which make up the function, to find the inverse: _________________________________________. Example: f: {(3,7) , (4,2) , (5,-3) , (6,-9)} Inverse of f: ____________________________. B. If you know the graph of a function, to find the graph of the inverse, ___________________________________________________. Examples: C. If you know the equation of a function, to find its inverse: ___________________________________________________. Example: f(x) = Y = 3x2 – 2x + 5 k(x) = Y = 2x – 3 (more on this in the 8.3 notes) Inverse of f: __________________ Inverse of k: __________________ Algebra 2 Notes Ch. 8 – Inverses & Radicals p. 4 of 6 III. Under what conditions will the inverse of a function also be a function? A. The function must be _______. This means that each ___ must be paired up with exactly one ___ (and therefore it is a __________), and each ___ must be paired up with exactly one ___ (that’s the 1-1 part). Ex. {(2,4) , (5,7) , (1,-3) , (7,4)} Function _____; 1-1 _____ {(2,4) , (5,7) , (2,-3) , (7,-4)} Function _____; 1-1 _____ {(2,4) , (5,7) , (1,-3) , (7,-4)} Function _____; 1-1 _____ B. If the graph is given, apply both the Vertical and the Horizontal Line Tests. The Vertical Line Test to show the original is a function. The Horizontal Line Test is applied to the __________ to show the __________ is a function. Examples: Algebra 2 Notes Ch. 8 – Inverses & Radicals p. 5 of 6 8.3 – INVERSES OF RELATIONS I. Notation f –1 is called the “inverse of the function f” and it is read “f inverse.” Steps to finding an Inverse: 1. Substitute y for f(x) 2. Exchange the y and x 3. Solve back for y 4. Substitute f –1 (x) for y. Given: f(x) = 3x + 5 Find f –1(x) Given: f(x) = x – 9 4 Find f –1(x) Given: f(x) = 5x7 + 4 Find f –1(x) Algebra 2 Notes Ch. 8 – Inverses & Radicals p. 6 of 6 II. Working with inverses Given: f(x) = 3x + 5 and f –1(x) = x – 5 3 Find f(f –1(3)) = Find f –1(f(5)) = Find f –1(f(x)) = Find f(f –1(x)) = If f and g are functions such that f(g(x)) = ___ and g(f(x)) = ___ for all x, then f and g are inverse functions