Understanding Inverse Functions in Mathematics

Chabot Mathematics §9.2bInverse Fcns Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

MTH 55 9.2 Review § • Any QUESTIONS About • §9.2 → Composite Functions • Any QUESTIONS About HomeWork • §9.2 → HW-43

Toys States Domain Range (inputs) (outputs) Domain Range (inputs) (outputs) Maine 1 Illinois 7 Iowa 2 Ohio 3 ball Ann rope Jim phone Jack car Inverse & One-to-One Functions • Let’s view the following two functions as relations, or correspondences:

Toys States Range Domain (inputs) (outputs) Range Domain (inputs) (outputs) Maine 1 Illinois 7 Iowa 2 Ohio 3 ball Ann rope Jim phone Jack Car Inverse & One-to-One Functions • Suppose we reverse the arrows. We obtain what is called the inverse relation. Are these inverse relations functions?

Toys States Range Domain (inputs) (outputs) Range Domain (inputs) (outputs) Maine 1 Illinois 7 Iowa 2 Ohio 3 ball Ann rope Jim phone Jack Car Inverse & One-to-One Functions • Recall that for each input, a function provides exactly one output. The inverse of “States” correspondence IS a function, but the inverse of “Toys” is NOT.

One-to-One for “States” Fcn • In the States function, different inputs have different outputs, so it is a one-to-one function. • In the Toys function, rope and phone are both paired with Jim. • Thus the Toy function is NOT one-to-one.

One-to-One Summarized • A function f is one-to-one if different inputs have different outputs. That is, if for a and b in the domain of f with a≠ b we have f(a) ≠ f(b) then the function f is one-to-one. • If a function is one-to-one, then its INVERSE correspondence is ALSO a FUNCTION.

One-to-One Fcn Graphically • Each y-value in the range corresponds to only one x-value in the domain • i.e.; Each x has a Uniquey

NOT a One-to-One Fcn • The y-value y2 in the range corresponds to TWO x-values, x2 and x3, in the domain.

NOT a Function at All • The x-value x2 in the domain corresponds to the TWO y-values, y2 and y3, in the range.

Definition of Inverse Function • Let f represent a one-to-one function. The inverse of f is also a function, called the inverse function of f, and is denoted by f−1. • If (x, y) is an ordered pair of f, then (y, x) is an ordered pair of f−1, and we write x = f−1(y). We have y = f (x) if and only if f−1(y) = x.

Example  f-values ↔ f-1-values • Assume that f is a one-to-one function. • If f(3) = 5, find f-1(5) • If f-1(−1) = 7, find f(7) • Solution: Recall that y = f(x) if and only if f-1(y) = x • Let x = 3 and y = 5. Now 5 = f(3) if and only if f−1(5) = 3. Thus, f−1(5) = 3. • Let y = −1 and x = 7. Now, f−1(−1) = 7 if and only if f(7) = −1. Thus, f (7) = −1.

1. for every x in the domain of f–1. 2. for every x in the domain of f . Inverse Function Property • Let f denote a one-to-one function. Then

Example  Inverse Fcn Property • Let f(x) = x3 + 1. Show that • Soln:

for every x in the domain of g and for every x in the domain of f, then UNIQUE Inverse Fcn Property • Let f denote a one-to-one function. Then if g is any function such that g = f–1. That is, g is the inverse function of f.

Verify Inverse Functions • Verify that the following pairs of functions are inverses of each other: • Solution: From the composition of f & g.

Verify Inverse Functions • Solution (cont.): Now Find • Observe: • This Verifies that f and gare indeed inverses of each other.

Example  Find Inverse of a Fcn • Given that f(x) = 5x− 2 is one-to-one, then find an equation for its inverse • Solution: f (x) = 5x – 2 y = 5x – 2 Replace f(x) with y x = 5y – 2 Interchange x and y Solve for y Replace y with f-1(x)

Procedure for finding f−1 • Replace f(x) by y in the equation for f(x). • Interchange x and y. • Solve the equation in Step 2 for y. • Replace y with f−1(x).

Example  Find the Inverse • Find the inverse of the one-to-one function • Solution: Step 1 Step 2 Step 3

Example  Find the Inverse Step 3 (cont.) Step 4

Example  Find Domain & Range • Find the Domain &Range of the function • Solution: Domain of f, all real numbers x such that x ≠ 2, in interval notation (−∞, 2)U(2, −∞) • Range of f is the domain of f−1 • Range of f is (−∞, 1) U (1, −∞)

Inverse Function Machine • Let’s consider inverses of functions in terms of function machines. Suppose that a one-to-one function f, has been programmed into a machine. • If the machine has a reverse switch, when the switch is thrown, the machine performs the inverse function, f−1. Inputs then enter at the opposite end, and the entire process is reversed.

Reverse Switch Graphically Reverse Forward

Recall that to be a Function an (x,y) relation must pass the VERTICAL LINE test Horizontal Line Test • In order for a function to have an inverse that is a function, it must pass the HORIZONTAL-LINE test as well • NOT a Function – Fails the Vertical Line Test

If it is impossible to draw a horizontal line that intersects a function’s graph more than once, then the function isone-to-one. For every one-to-one function, an inverse function exists. Horizontal Line Test Defined • A Function withOUT and Inverse – Fails the Horizontal Line Test (not 1-to-1)

Example  Horizontal Line Test • Determine whether the function f(x) = x2 + 1 is one-to-one and thus has an inverse fcn. • The graph of f is shown. Many horizontal lines cross the graph more than once. For example, the line y = 2 crosses where the first coordinates are 1 and −1. Although they have different inputs, they have the same output: f(−1) = 2 = f(1). The function is NOT one-to-one, therefore NO inverse function exists y 8 7 6 5 4 3 2 1 -5 -4 -3 -2 -1 1 2 3 4 5 x -1 -2

Example  Horizontal Ln Test • Use the horizontal-line test to determine which of the following fcns are 1-to-1 a. b. • Soln a. • No horizontal line intersects the graph of f in more than one point, therefore the function fis one-to-one

Example  Horizontal Ln Test • Use the horizontal-line test to determine which of the following fcns are 1-to-1 a. b. • Soln b. • No horizontal line intersects the graph of f in more than one point, therefore the function f is 1-to-1

Graphing Fcns and Their Inverses • How do the graphs of a function and its inverse compare?

f (x) = 5x – 2 6 5 4 3 2 1 -5 -4 -3 -2 -1 1 2 3 4 5 x -1 -2 f-1(x) = (x + 2)/5 -3 -4 -5 Example  Graphs Inverse Fcn • Graph f(x) = 5x− 2 and f−1(x) = (x + 2)/5 on the same set of axes and compare • Solution: • Note that the graph of f−1(x) can be drawn by reflecting the graph of f across the line y = x. • When x and y are interchanged to find a formula for f−1(x), we are, in effect, Reflecting or Flipping the graph of f.

Visualizing Inverses • The graph of f−1is a REFLECTION of the graph of f across the line y = x.

Example  Use y = x Mirror Ln • The graph of the function f is shown at Lower Right. Sketch the graph of the f−1 • Soln

Example  Inverse or Not? • Ray’s Music Mart has six employees. The first table lists the first names and the Social Security numbers of the employees, and the second table lists the first names and the ages of the employees • Find the inverse of the function defined by the first table, and determine whether the inverse relation is a function • Find the inverse of the function defined by the second table, and determine whether the inverse relation is a function

Example  Inverse or Not? • Solution:Every y-value corresponds to exactly onex-value. Thus the inverse of the function defined in this table is a function

Example  Inverse or Not? • Solution:There is more than one x-value that corresponds to a y-value. For example, the age of 24 yields the names Dwayne and Anna. Thus the inverse of the function defined in this table is NOT a function.

Example  Hydrostatic Pressure • The formula for finding the water pressure p (in pounds per square inch, or psi),at a depth d (in feet) below the surface is • A pressure gauge on a Diving Bell breaks and shows a reading of 1800 psi. Determine how far below the surface the bell was when the gauge failed

Example  HydroStatic P • Solution: The depth is given by the inverse of • Solve theInverseEqn for p Let p = 1800 psi • The Diving Bell was 3960 feet below the surface when the gauge failed

WhiteBoard Work • Problems From §9.2 Exercise Set • 38, 42, 60, 68, 76 • Some Temperature Scales

All Done for Today Old StyleDivingBell

Chabot Mathematics Appendix Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu –

Understanding Inverse Functions in Mathematics