6.7 Notes – Inverse Functions. Notice how the x-y values are reversed for the original function...
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Transcript of 6.7 Notes – Inverse Functions. Notice how the x-y values are reversed for the original function...
Bellwork:On graph paper, sketch the following functions on the SAME set of axes:1) 2) 3) 4)
What do you notice about the graphs?
Homework:Read 6.76.7 Part 1 (1-3, 12-20, 42-47)6.7 Complete Solutions - Part 1 (in documents)
𝑦=√𝑥−2
𝑦=𝑥𝑦=𝑥2+2
𝑦=−√𝑥−2If you reflect the graph of across the line , you end up with the graph for !
Reflecting across line is the same as just swapping the x-values and y-values!
If you reflect the graph of across the line , you end up with the graph for !
Reflecting across line is the same as just swapping the x-values and y-values! 𝑦=𝑥2+2 𝑦=√𝑥−2 𝑦=−√𝑥−2
Notice how the x-y values are reversed for the original function and the reflected functions.
Definitions:Inverse Relations – Relations that UNDO eachother.
The graphs of inverse relations are reflections of each other across the line
The coordinates for x and y are swapped when going between a relation and its inverse.
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 = x2 x y
-2 4
-1 1
0 0
1 1
Equation
Table of values
Graph
• Inverse relation – just think: switch the x & y-values.
x = y2
xy
x y
4 -2
5 -1
0 0
1 1
** 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
yx
3
6
23
1 xy 2
3
11 xy
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)=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(x+2)+6
= x-6+6
= x
g(f(x))= (-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.
4. Call y what it actually is: .
** Remember functions have to pass the vertical line test!
Ex: (a)Find the inverse of f(x)=x5.
1. y = x5
2. x = y5
3. 5 55 yx
yx 5
5 xy
(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.
5 x
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)=2x2-4 Determine whether f -1(x) is a function, then find the inverse equation.
2
2
4y
x
f -1(x)= is not a function.
y = 2x2-4
x = 2y2-4
x+4 = 2y2
2
4x
y
22
1 xyOR, if you fix the
tent in the basement…
22
1 x