INVERSE FUNCTIONS Remember we talked about functionstaking a
INVERSE FUNCTIONS
Remember we talked about 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
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
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
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:
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