Function Compositions and Inverses Function Notation fx does

Function Compositions and Inverses

Function Notation • f(x) does NOT stand for MULTIPLICATION! • We read f(x) “f of x” and it means that the function’s independent variable is x. • If the function is defined as f(x) = 3 x - 1 and we are asked to find f(2), we just substitute 2 for x in the function: f(2) = 3(2) - 1 = 6 - 1 = 5

Function Substitutions 2 x Given g(x) = - x, find g(-3) Substitute -3 for x: 2 g(-3) = (-3) - (-3) = 12 g(-3) = 12

Function Substitutions 2 4 x Given g(x) = 3 x + 2, find g(5) Substitute 5 for x 2 g(5) = 3(5) - 4(5) + 2 = -83 g(5) = -83

Function Composition is just more substitution, very similar to what we have been doing with finding the value of a function. The difference is we will be substituting another function instead of a number. . .

Function Composition For example… Given f(x) = x - 5, find f(a+1) Substitute (a+1) for x f(a + 1) = (a + 1) - 5 = a+1 - 5 The answer is a function in terms of ‘a’ =a-4

Function Composition • Composition notation looks like g(f(x)) or f(g(x)), we read this ‘g of f of x’ or ‘f of g of x’. • We are given f(x) and g(x), the function inside the parentheses gets substituted into the other.

Function Composition Given the functions: f(x) = 2 x+2 & g(x) = 2 find f(g(x)) This notation tells us to substitute the g(x) function, 2, for x in the f(x) function: f (2) = 2(2)+2 =6

Function Composition Given the functions: g(x) = x - 5 & f(x) = x + 1 find f(g(x)) This notation tells us to substitute the g(x) function, x-5, for x in the f(x) function: f (x-5) = (x-5)+1 =x-4

Function Composition Reverse the composition: g(x) = x - 5 & f(x) = x + 1 find g(f(x)) This notation tells us to substitute the f(x) function, x+1, for x in the g(x) function: g(x+1) = (x+1)-5 =x-4

Function Composition In the last example, f(g(x)) and g(f(x)) had the same results. This is not always the case. Try this example: f(x) = x 2 + x & g(x) = x - 4 find f(g(x)) and g(f(x))

Function Composition f(x) = 2 x + x & g(x) = x - 4 1) f(g(x)) = f(x-4) = (x-4)2 + (x-4) = x 2 -8 x+16+x-4 = x 2 -7 x+12 2) g(f(x)) = g(x 2 + x ) = (x 2 + x )-4 = x 2 + x - 4

Function Composition New Example: Given f(x) = 2 x + 5 & g(x) = 8 + x find f(g(-5) & g(f(-5) 1) f(g(-5) : find g(-5) = 8 + (-5) = 3 then find f(3) = 2(3) + 5 = 11 2) g(f(-5)) : find f(-5) = 2(-5) + 5 = -5 then find g(-5) = 8 + (-5) = 3

Function Inverse Remember: a function is a set of ordered pairs (including lists of discrete points and also equations which give us infinite points), where no two points have the same xcoordinate. The Inverse of a function is the set of points where each point in the function is reversed, (y, x).

Function Inverse A function that is a list of ordered pairs is easy to find the inverse of: f(x) = {(1, 2), (2, 5), (3, -4), (4, 0)} The inverse is: f-1(x) = {(2, 1), (5, 2), (-4, 3), (0, 4)}

Function Inverse To find the inverse of a function that is written as an equation, like: f(x) = x + 7 We will: 1) Replace the function label, f(x) with y 2) Swap the variables, x and y 3) Solve the new equation for y

Function Inverse Find the Inverse of: f(x) = x + 7 Replace: y=x+7 Swap: x = y + 7 Solve: y = x - 7 f-1(x) = x - 7

Function Inverse Find the Inverse of: f(x) = 3 x - 4 Replace: y = 3 x - 4 Swap: x = 3 y - 4 Solve: 3 y = x + 4 y = (x + 4)/3 f-1(x) = (x + 4)/3

Function Inverse Find the Inverse of: f(x) = (2 x + 5)/3 Replace: y = (2 x + 5)/3 Swap: x = (2 y + 5)/3 Solve: 3 x = 2 y + 5 2 y = 3 x - 5 f-1(x) = (3 x - 5)/2

Function Inverse Find the Inverse of: f(x) = x 2 - 4 Replace: y = x 2 - 4 Swap: x = y 2 - 4 Solve: y 2 = x + 4 y = ±√x + 4

Function Inverse Note: In the last example, the inverse does NOT pass the test to be a function. This sometimes happens, that the inverse of a function is not a function. This occurs when the function has points with the same y-value (allowed in functions).

Function Inverse A function whose inverse IS ALSO a function is called a ONE-TO-ONE function. Each x-coordinate has a different ycoordinate and each y-coordinate has a different x-coordinate.