Presentation on theme: "Inverse Relations Objectives: Students will be able to…"— Presentation transcript:
1Inverse Relations Objectives: Students will be able to… Determine whether a function has an inverseWrite an inverse functionVerify 2 functions are inverses of each otherInverse Relations
2The inverse of a given function will “undo” what the original function did. For example, let’s take a look at the square function: f(x) = x2xyf-1(x)f(x)933993399339933x299x3399933399
3Inverse Relations: The inverse of f is f -1 (read “f inverse”) If both the original relation and the inverse relation are both functions, they are inverse functions!The domain of the original relation is the range of the inverse.Inverse Relations:
4Graphically, the x and y values of a function and its inverse are switched.If the function y = g(x) contains the pointsx1234y816then its inverse, y = g-1(x), contains the pointsx124816y3Where is there a line of reflection?
5y = f(x)y = xThe graph of a function and its inverse are mirror images about the liney = f-1(x)y = x
6Find the inverse relation: x-3-15710y-54xyFind the inverse relation:
7Writing the equation of an inverse function: Example 1: y = 6x - 12Step 1: Switch x and y:x = 6y - 12Step 2: Solve for y:
8Find an equation for the inverse relation: 1.2.Find an equation for the inverse relation:
9Verifying 2 functions are inverses If f and g are inverse functions, their composition would simply give x back.Verifying 2 functions are inverses
10Ex: 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.Because f(g(x))=x and g(f(x))=x, they are inverses.f(g(x))= -3(-1/3x+2)+6= x-6+6= xg(f(x))= -1/3(-3x+6)+2= x-2+2= x
11Verify that the functions f and g are inverses of each other. Since both of these = x, if you start with x and apply the functions they “undo” each other and are inverses.
12Remember…The graph of a function needs to pass the vertical line test in order to be a function.
13Used 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 will pass the vertical line test and therefore is a function.If the original function does not pass the horizontal line test, then its inverse is not a function (may need to restrict domain so that it has an inverse).Horizontal Line Test
14All functions have only one y value for any given x value (this means they pass the vertical line test)One-to-one functions also have only one x- value for any given y-value.One to One Functions
15If a horizontal line intersects the graph of an equation more than one time, the equation graphed is NOT a one-to-one function and will NOT have an inverse function.This is a one-to-one functionThis is NOT a one-to-one functionThis is NOT a one-to-one function
16Graph does not pass the horizontal line test, therefore the inverse is not a function. Ex: Graph the function f(x)=x2 and determine whether its inverse is a function.
17But what if I restricted the domain of x2 But what if I restricted the domain of x2? Then it would have an inverse.Graph y = x2 for x > 0. Find the inverse function.
18Ex: Graph f(x)=2x2-4. Determine whether f -1(x) is a function. How could I restrict the domain to make it have an inverse function?x > 0f -1(x) is not a function.
19Example 2: Given the function : y = 2x2 -4, x > 0 find the inverse Step 1: Switch x and y:x = 2y2 -4Step 2: Solve for y:Only need the positive root for inverse!
20Ex: Write the inverse of g(x)=2x3 y=2x3x=2y3Inverse is a function!
21Graph the functions to determine whether their inverses will also be functions 𝑦= 𝑥 2 +8𝑦=± 𝑥 2 −5𝑦=− 𝑥 4 − 𝑥 2 +4𝑥−3𝑦= 𝑥 2 +4𝑥−2, 𝑥≥−2𝑦= 𝑥 3 +7 𝑥 2 −10𝑦= sin 𝑥