Imperial Valley College Math Lab Functions: Composition and Inverse Functions FUNCTION COMPOSITION In order to perform a composition of functions, it is essential to be familiar with function notation. If you see something of the form “푓(푥) = [expression in terms of x]”, this means that whatever you see in the parentheses following f should be substituted for x in the expression. This can include numbers, variables, other expressions, and even other functions. EXAMPLE: 푓(푥) = 4푥2 − 13푥 푓(2) = 4 ∙ 22 − 13(2) 푓(−9) = 4(−9)2 − 13(−9) 푓(푎) = 4푎2 − 13푎 푓(푐3) = 4(푐3)2 − 13푐3 푓(ℎ + 5) = 4(ℎ + 5)2 − 13(ℎ + 5) Etc. A composition of functions occurs when one function is “plugged into” another function. The notation (푓 ○푔)(푥) is pronounced “푓 of 푔 of 푥”, and it literally means 푓(푔(푥)). In other words, you “plug” the 푔(푥) function into the 푓(푥) function. Similarly, (푔 ○푓)(푥) is pronounced “푔 of 푓 of 푥”, and it literally means 푔(푓(푥)). In other words, you “plug” the 푓(푥) function into the 푔(푥) function. WARNING: Be careful not to confuse (푓 ○푔)(푥) with (푓 ∙ 푔)(푥), which means 푓(푥) ∙ 푔(푥) . EXAMPLES: 푓(푥) = 4푥2 − 13푥 푔(푥) = 2푥 + 1 a. (푓 ○푔)(푥) = 푓(푔(푥)) = 4[푔(푥)]2 − 13 ∙ 푔(푥) = 4(2푥 + 1)2 − 13(2푥 + 1) = [푠푚푝푙푓푦] … = 16푥2 − 10푥 − 9 b. (푔 ○푓)(푥) = 푔(푓(푥)) = 2 ∙ 푓(푥) + 1 = 2(4푥2 − 13푥) + 1 = 8푥2 − 26푥 + 1 A function can even be “composed” with itself: c. (푔 ○푔)(푥) = 푔(푔(푥)) = 2 ∙ 푔(푥) + 1 = 2(2푥 + 1) + 1 = 4푥 + 3 INVERSE FUNCTIONS ퟏ The notation for inverse functions can cause confusion. It is important to know that 풇−ퟏ(풙) ≠ . 풇(풙) Instead, 푓−1(푥) indicates the inverse function of 풇(풙), which can be thought of as the function that “reverses” 푓(푥) , or “undoes” everything that 푓(푥) does. EXAMPLE: Let 푓(푥) = 3푥 − 1 . In words, 푓(푥) takes a number, multiplies it by 3, and then subtracts 1 from it. The opposite or reverse of this procedure would be to add 1 to a number, then divide it by 3. For instance, 푓(ퟒ) = 3 ∙ 4 − 1 = ퟏퟏ . Input = 4 , Output = 11 In reverse, take the output 11, add 1 to it, then divide by 3: 11 + 1 = 12 12 ÷ 3 = ퟒ = input . In other words, the output became the input, and vice-versa. They switched roles! Mathematically, In each ordered pair (푥, 푦) associated with a function, the x and y “switch places”: (푥, 푦) → (푦, 푥) In the above example, (4, 11) became (11, 4) . Procedure: EXAMPLE: Given 푓(푥) , take the following steps to find 푓−1(푥): 1) 푦 = 3푥 − 1 1) Replace 푓(푥) with y in the equation 2) 푥 = 3푦 − 1 2) Swap the x and y in the equation (they “trade places”) 푥+1 3) Solve the equation for y (i.e., isolate the y) 3) 푥 + 1 = 3푦 → = 푦 3 4) Once y is isolated, replace it with 푓−1(푥) . 푥+1 1 1 4) 푓−1(푥) = or 푓−1(푥) = 푥 + Check that these values work in both directions: 3 3 3 푓(푥) = 3푥 − 1 y ( ) 1) ퟒ, ퟏퟏ –5 –16 2) (풙, 풚) (ퟏퟏ, ퟒ) 1 2 3) (풚, 풙) −1 x 4 11 푓 (푥) −1 푥+1 4) 푓 (푥) = 3 푓(푥) −1 푥+1 푓 (푥) = 3 Notice how the graphs of 풇(풙) and 풇−ퟏ(풙) are symmetric (mirror images) across the diagonal line y = x, which is the result of all the x- and y-values “trading places”. Imagine each (푥, 푦) point as “hopping” across the line y = x, becoming (푦, 푥). .