3 Inverse Functions

Relation – a mapping of input values (x-values) onto output values (y-values). Here are 3 ways to show the same relation. x y -2 4 -1 1 0 0 1 1 y = x2 Equation Table of values Graph

Inverse relation – just think: switch the x & y-values. -2 -1 0 0 1 1 x = y2 ** the inverse of an equation: switch the x & y and solve for y. ** the inverse of a table: switch the x & y. ** the inverse of a graph: the reflection of the original graph in the line y = x.

Ex: Find an inverse of y = -3x+6. Steps: -switch x & y -solve for y y = -3x+6 x = -3y+6 x-6 = -3y

Inverse Functions Given 2 functions, f(x) & g(x), if f(g(x))=x AND g(f(x))=x, then f(x) & g(x) are inverses of each other. Symbols: f -1(x) means “f inverse of x”

Ex: Verify that f(x)=-3x+6 and g(x)=-1/3x+2 are inverses. Meaning find f(g(x)) and g(f(x)). If they both equal x, then they are inverses. f(g(x))= -3(-1/3x+2)+6 = x-6+6 = x g(f(x))= -1/3(-3x+6)+2 = x-2+2 = x ** Because f(g(x))=x and g(f(x))=x, they are inverses.

To find the inverse of a function: Switch the x & y values. Solve the new equation for y. ** Remember functions have to pass the vertical line test!

Ex: (a)Find the inverse of f(x)=x5. (b) Is f -1(x) a function? (hint: look at the graph! Does it pass the vertical line test?) y = x5 x = y5 Yes , f -1(x) is a function.

Horizontal Line Test Used to determine whether a function’s inverse will be a function by seeing if the original function passes the horizontal line test. If the original function passes the horizontal line test, then its inverse is a function. If the original function does not pass the horizontal line test, then its inverse is not a function.

Ex: Graph the function f(x)=x2 and determine whether its inverse is a function. Graph does not pass the horizontal line test, therefore the inverse is not a function.

Ex: f(x)=7x-1 Determine whether f -1(x) is a function, then find the inverse equation. y = 7x - 1 x = 7y- 1 x+ 1 = 7y f -1(x) is a function.