7.4 Inverse Functions p. 422 What is an inverse relation? What do you switch to find an inverse relation? What notation is used for an inverse function?

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Presentation transcript:

7.4 Inverse Functions p. 422 What is an inverse relation? What do you switch to find an inverse relation? What notation is used for an inverse function? How is it read? What test can you use to verify the inverse of a function is a function?

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

Inverse relation – just think: switch the x & y-values. x = y 2 x y ** 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 / 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))= -3(- 1 / 3 x+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: 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!

Ex: (a)Find the inverse of f(x)=x 5. 1.y = x 5 2.x = y 5 3. (b) Is f -1 (x) a function? (hint: look at the graph! Does it pass the vertical line test?) 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 h hh horizontal line test. If the original function p pp passes the horizontal line test, then its i ii inverse is a function. If the original function d dd does not pass the horizontal line test, then its i ii 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.

Ex: f(x)=2x 2 -4 Determine whether f -1 (x) is a function, then find the inverse equation. f -1 (x) is not a function. y = 2x 2 -4 x = 2y 2 -4 x+4 = 2y 2 OR, if you fix the tent in the basement…

Ex: g(x)=2x 3 Inverse is a function! y=2x 3 x=2y 3 OR, if you fix the tent in the basement…

What is an inverse relation?What is an inverse relation? It maps the output values back to the original input values (domain or inverse is range or original). What do you switch to find an inverse relation?What do you switch to find an inverse relation? x and y What notation is used for an inverse function?What notation is used for an inverse function? f -1 (x) How is it read?How is it read? f inverse of x What test can you use to verify the inverse of a function is a function?What test can you use to verify the inverse of a function is a function? Horizontal line test

Assignment Page 426, odd