INVERSE FUNCTIONS Remember functionstaking a set X and
INVERSE FUNCTIONS
Remember functions--taking a set X and mapping into a Set Y 11 22 33 44 55 Set X 22 44 66 88 1010 Set Y An inverse function would reverse that process and map from Set. Y back into Set X
If we map what we get out of the function back, we won’t always have a function going back!!! 1 2 3 4 5 2 4 6 8
Inverse Relations The inverse of a relation is the set of ordered pairs obtained by switching the input with the output of each ordered pair in the original relation. (The domain of the original is the range of the inverse; and vice versa) Ex: and are inverses because their input and output are switched. For instance: 22 4
Tables and Graphs of Inverses Switch x and y Orginal (0, 25) (20, 25) (18, 16) (2, 16) (6, 4) (14, 4) (10, 0) X 0 2 6 10 14 18 20 Y 25 16 4 0 4 16 25 X 25 16 4 0 4 16 25 Y 0 2 6 10 14 18 20 Inverse Switch x and y (16, 18) (4, 14) (0, 10) (4, 6) (16, 2) Although transformed, the graphs are identical Line of Symmetry: y = x
Inverse and Compositions In order for two functions to be inverses: AND
Recall that to determine by the graph if an equation is a function, we have the vertical line test. If a vertical line intersects the graph of an equation more than one time, the equation graphed is NOT a function. This is a function This is NOT a function This is a function
If the inverse is a function, each y value could only be paired with one x. Let’s look at a couple of graphs. Look at a y value (for example y = 3)and see if there is only one x value on the graph for it. For any y value, a horizontal line will only intersection the graph once so will only have one x value Horizontal Line Test to see if the inverse is a function.
If 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 function This is NOT a one -to-one function
The Horizontal Line Test If a horizontal line intersects a curve more than once, it’s inverse is not a function. Use the horizontal line test to decide which graphs have an inverse that is a function. Make sure to circle the functions.
The Horizontal Line Test If a horizontal line intersects a curve more than once, it’s inverse is not a function. Use the horizontal line test to decide which graphs have an inverse that is a function. Make sure to circle the functions.
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
One-to-One Functions A function f(x) is one-to-one on a domain D if, for every value c, the equation f(x) = c has at most one solution for every x in D. Or, for every a and b in D: Theorems: 1. A function has an inverse function if and only if it is one-to -one. 2. If f is strictly monotonic (strictly increasing or decreasing) on its entire domain, then it is one-to-one and therefore has an inverse function.
Steps for Finding the Inverse of a One-to-One Function y = f -1(x) Solve for y Trade x and y places Replace f(x) with y
Let’s check this by doing Find the inverse of y = f -1(x) or Solve for y Yes! Trade x and y places Replace f(x) with y Ensure f(x) is one to one first. Domain may need to be restricted.
Find the inverse of a function : Example 1: y = 6 x - 12 Step 1: Switch x and y: x = 6 y - 12 Step 2: Solve for y:
Example 2: Given the function : y = 3 x 2 + 2 find the inverse: Step 1: Switch x and y: x = 3 y 2 + 2 Step 2: Solve for y:
Ex: Find an inverse of y = -3 x+6. • Steps: -switch x & y -solve for y y = -3 x+6 x = -3 y+6 x-6 = -3 y
Finding the Inverse Try
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 Equation Table of values Graph x y -2 4 -1 1 0 0 1 1
• Inverse relation – just think: switch the x & y-values. x = y 2 ** the inverse of an equation: switch the x & y and solve for y. x y 4 -2 1 -1 0 0 1 1 ** 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.
Consider the graph of the function The inverse function is
Consider the graph of the function x x The inverse function is An inverse function is just a rearrangement with x and y swapped. So the graphs just swap x and y!
What else do you notice about the graphs? x x x is a reflection of in the line y = x The function and its inverse must meet on y = x
Graph f(x) and f -1(x) on the same graph. -4 0 4 5 3 5 -4 0 4
Graph f(x) and f -1(x) on the same graph. -4 -4 -3 -5 0 4 -4 -5 4 -4 -3 0
Notice that the x and y values Let’s consider the function traded places for the function some values graph them. and its and inverse. x -2 -1 0 1 2 f (x) -8 -1 0 1 8 These functions are and compute reflections of each other about the line y = x This means “inverse function” (2, 8) (8, 2) x -8 -1 Let’s take the 0 values we got out of the function and 1 put them into the 8 f -1(x) -2 -1 0 1 2 (-8, -2) (-2, -8) inverse function and plot them Is this a function? What will “undo” a cube? Yes A cube root
Graph f(x) = 3 x − 2 and using the same set of axes. Then compare the two graphs. Determine the domain and range of the function and its inverse.
Verify that the functions f and g are inverses of each other. If we graph (x - 2)2 it is a parabola shifted right 2. Is this a one-to-one function? This would not be one-to-one but they restricted the domain and are only taking the function where x is greater than or equal to 2 so we will have a one-to-one function.
e. g. On the same axes, sketch the graph of and its inverse. Solution: x
Ex: f(x)=2 x 2 -4 Determine whether f -1(x) is a function, then find the inverse equation. y = 2 x 2 -4 x = 2 y 2 -4 x+4 = 2 y 2 f -1(x) is not a function. OR, if you fix the tent in the
Ex: g(x)=2 x 3 y=2 x 3 x=2 y 3 Inverse is a function! OR, if you fix the tent in the basement…
Exercise 1 (a) Sketch the function where. (b) Write down the range of . (c) Suggest a suitable domain for so that the inverse function can be found. (d) Find and write down its domain and range. (e) On the same axes sketch .
Solution: (b) Range of (a) : (c) Restricted domain: ( We’ll look at the other possibility in a minute. ) (d) Inverse: Let Rearrange: Swap: Domain: Range:
Solution: (b) Range of (a) (c) : Suppose you chose for the domain (d)As before Let Rearrange: We now need since
Solution: (a) (b) Range: (c) for the domain Choosing is easier! (d)As before Let Rearrange: We now need Swap: Domain: Suppose you chose since Range:
Inverse Trigonometry
Cosine Tangent Each one of these trigonometric functions fail the horizontal line test, so they are not one-to-one. Therefore, there inverses are not functions. Cosecant Secant Cotangent Sine Trigonometric Functions
Cosine Tangent In order for their inverses to be functions, the domains of the trigonometric functions are restricted so that they become one-toone. Cosecant Secant Cotangent Sine Trigonometric Functions with Restricted Domains
Trigonometric Functions with Restricted Domains Function f (x) = sin x f (x) = cos x f (x) = tan x f (x) = csc x f (x) = sec x f (x) = cot x Domain Range
Cos-1 Sec-1 Cot-1 Tan-1 Csc-1 Sin-1 Inverse Trigonometric Functions
Inverse Trigonometric Functions Function f (x) = sin-1 x f (x) = cos-1 x -1 f (x) = tan x f (x) = csc-1 x f (x) = sec-1 x -1 f (x) = cot x Domain Range
Alternate Names/Defintions for Inverse Trigonometric Functions Familiar f (x) = sin-1 x -1 f (x) = cos x -1 f (x) = tan x f (x) = csc-1 x -1 f (x) = sec x -1 f (x) = cot x Alternate Calculator f (x) = arcsin x f (x) = sin-1 x -1 f (x) = arccos x f (x) = cos x -1 f (x) = arctan x f (x) = arccsc x f (x) = sin-1 1/x -1 f (x) = arcsec x f (x) = cos 1/x -1 f (x) = arccot x f (x) = -tan x+ Arccot is different because it is always positive but tan can be negative.
Example 1 Evaluate: This expression asks us to find the angle whose sine is ½. Remember the range of the inverse of sine is.
Example 2 Evaluate: This expression asks us to find the angle whose cosecant is -1 (or sine is -1). Remember the range of the inverse of cosecant is.
Example 3 Evaluate: The embedded expression asks us to find the angle whose sine is 1/3. Draw a picture (There are infinite varieties): It does not even matter what the angle is, we only need to find: Find the missing side length(s) Is the result positive or negative?
Example 4 Evaluate: The embedded expression asks us to find the angle whose cosine is -1/6. Ignore the Draw a picture (There are infinite varieties): negative for now. It does not even matter what the angle is, we only need to find: Find the missing side length(s) Is the result positive or negative?
Example 3 Evaluate: The embedded expression asks us to find the angle whose tangent is x. Draw a possible picture (There are infinite varieties): It does not even matter what the angle is, we only need to find: Find the missing side length(s) Is the result positive or negative?
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