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) = 3x - 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 Given g(x) = x2 - x, find g(-3) Substitute -3 for x: g(-3) = (-3)2 - (-3) = 12 g(-3) = 12

Function Substitutions Given g(x) = 3x - 4x2 + 2, find g(5) Substitute 5 for x g(5) = 3(5) - 4(5)2 + 2 = -83 g(5) = -83

Function Composition 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 = a - 4 The answer is a function in terms of ‘a’

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) = 2x+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) = x2 + x & g(x) = x - 4 find f(g(x)) and g(f(x))

Function Composition f(x) = x2 + x & g(x) = x - 4 1) f(g(x)) = f(x-4) = (x-4)2 + (x-4) = x2 -8x+16+x-4 = x2 -7x+12 2) g(f(x)) = g(x2 + x ) = (x2 + x )-4 = x2 + x - 4

Function Composition New Example: Given f(x) = 2x + 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 x-coordinate. 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) = 3x - 4 Replace: y = 3x - 4 Swap: x = 3y - 4 Solve: 3y = x + 4 y = (x + 4)/3 f-1(x) = (x + 4)/3

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

Function Inverse Find the Inverse of: f(x) = x2 - 4 Replace: y = x2 - 4 Swap: x = y2 - 4 Solve: y2 = 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 y-coordinate and each y-coordinate has a different x-coordinate.