Inverse of a Function Section 10.4 pages 567-574

Goals: Find inverses of relations Explore inverses of functions Find inverses of functions algebraically Find inverses of nonlinear functions An Inverse relation switches the input and output values of the original relation. Original ordered pair is (a, b), the inverse is. (b, a) Example: apples in; lemons out to lemons in; apples out

Finding Inverses of Relations Original set of points: Inverse set of points:

Write a formula for the input of the function: y = 5x - 3 Remember: x is the input y is the output Solve for x by: Undo the add or subtract Undo the multiply or divide 𝑦=5𝑥 −3 𝑦+3 =5𝑥 𝑦+3 5 = x

Find Inverse of a function algebraically Step 1: Set y equal to f(x). In other words replace f(x) with y Step 2: Switch x and y in the equations Step 3: Solve the equation for y Example: Find the inverse of f(x) = 3x – 5 y = 3x – 5 x = 3y – 5 x + 5 = 3y 𝑥+5 3 = y. Or you can write it as the inverse of 𝑓 𝑖𝑠 𝑔 𝑥 = 1 3 𝑥+ 5 3 Notation: Function is 𝑓 𝑥 Inverse function is 𝑓 −1 (𝑥)

Try it: Find the inverse of 𝑓 𝑥 = 1 4 x + 9 4 𝑓 𝑥 = 1 4 x + 9 4 y = 1 4 x + 9 4 𝑥 = 1 4 𝑦 + 9 4 𝑥 − 9 4 = 1 4 𝑦 4(𝑥 − 9 4 ) = 4( 1 4 𝑦) 4𝑥−9=𝑦 𝑜𝑟 𝑡ℎ𝑒 𝑖𝑛𝑣𝑒𝑟𝑠𝑒 𝑜𝑓 𝑓 𝑖𝑠 𝑓 −1 𝑥 =4𝑥−9

Functions Each x value has only one unique y-value. In other words, x can’t repeat. Example: (3, 2) and (3, -9) is NOT a function because x-value of 3 has two different y values Example: (5, 2) and (8, 2) is a function, because each x-value has a y-value, y can repeat. Vertical line test: on the graph, if you draw a vertical line and it intersects the graph in only one point it is a function. Horizontal line test: To test if a function will have an inverse that is also a function, apply a horizontal line test to the original function to see if it intersects the original graph in only on point. On the graph at the right: 𝑦=𝑥 2 −2𝑥 −3 𝑖𝑠 𝑎 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑟𝑒𝑑 𝑙𝑖𝑛𝑒 𝑖𝑛𝑡𝑒𝑟𝑠𝑒𝑐𝑡𝑠 𝑜𝑛𝑐𝑒 . However the inverse of this equation would not be a function unless the domain is restricted to only the positive numbers, because the green line interests the graph at two points.

Inverse of a Quadratic Function Finding the Inverse of a quadratic function is the same as linear functions with the exception of limiting the domain to only positive real numbers. Step 1: Set y equal to f(x). In other words replace f(x) with y Step 2: Switch x and y in the equations Step 3: Solve the equation for y Example: Find the inverse of the function: 𝑓 𝑥 =4 𝑥 2 +3 𝑓𝑜𝑟 𝑥 ≥0 𝑦=4𝑥 2 +3 𝑥=4𝑦 2 +3 𝑥−3=4𝑦 2 𝑥−3 4 = 𝑦 2 𝑥−3 4 = 𝑦 2 𝑥−3 2 = y; Inverse is a function only if 𝑥 ≥0;

Graph the function and it’s inverse Use the domain of only x ≥ 0 𝑓 𝑥 =4 𝑥 2 +3 g 𝑥 = 𝑥−3 2 y = x x f(x) 3 1 7 2 19 x f(x) 3 4 0.5 7 1 12 1.5 19 2 Notice: a function and its inverse are symmetrical about a line y = x or in other words, the line y = x is like a mirror reflecting the function and its inverse.

Homework: We will start this assignment in class on Wednesday …. Do page 572 #1, 2, 6, 7, 10-22 even Due on Thursday, March 28 We will start this assignment in class on Thursday … Do page 573 #30-44 even, 46-52 all Due on Friday, March 29