Objectives To find the inverse of functions algebraically
Objectives: To find the inverse of functions algebraically and graphically. To determine if relations are inverses and/or functions. Relation – a mapping of input values (xvalues) onto output values (y-values). Here are 3 ways to show the same relation: y= x 2 x y -2 4 Equation -1 1 Table of values 0 0 Graph 1 1 2 4
• To find an inverse of a relation, switch the x’s & y’s** y = x 2 x = y 2 x -2 -1 0 y ** Thatx 4 could be 4 in an 1 equation: 1 switch the 0 x & y, then 0 solve for y. 1 1 2 4 1 y -2 -1 ** Or in a table: switch the x&y values. 0 1 ** Or 4 on a 2 graph: reverse the coordinates. (It’s a reflection of the original graph over the line y = x. )
Ex: Find an inverse of y = -3 x+6. • Steps: Switch x & y. Solve for y. y = -3 x+6 x = -3 y+6 -6 -6 x-6 = -3 y
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)=-3 x+6 and g(x)=-1/3 x+2 are inverses. • Meaning find f(g(x)) and g(f(x)). If they both equal x, then they are inverses. f(g(x))=f(-1/3 x+2) g(f(x))=g(-3 x+6) =-3(-1/3 x+2)+6 =-1/3(-3 x+6)+2 = x-6+6 = x-2+2 =x =x ** Because f(g(x))=x and g(f(x))=x, they are inverses.
To find the inverse of a function: 1. Change the f(x) to a y. 2. Switch the x & y values. 3. Solve the new equation for y. ** Remember functions have to pass the vertical line test!
Horizontal Line Test • 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)=x 2 and determine whether its inverse is a function. Graph does not pass the horizontal line test, therefore the inverse is not a function.
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