Inverse Functions

Definition A function is a set of ordered pairs with no two first elements alike. f(x) = { (x,y) : (3, 2), (1, 4), (7, 6), (9,12) } But... what if we reverse the order of the pairs? This is also a function... it is the inverse function f -1 (x) = { (x,y) : (2, 3), (4, 1), (6, 7), (12, 9) }

Example (Smart Balan. Test) Consider an element of an electrical circuit which increases its resistance as a function of temperature. T = TempR = Resistance R = f(T)

Example We could also take the view that we wish to determine T, temperature as a function of R, resistance. R = ResistanceT = Temp T = g(R) Now we would say that g(R) and f(T) are inverse functions

Terminology If R = f(T)... resistance is a function of temperature, Then T = f -1 (R)... temperature is the inverse function of resistance. f -1 (R) is read "function-inverse of R“ is not an exponent it does not mean reciprocal

Does This Have An Inverse? Given the function at the right Can it have an inverse? Why or Why Not? NO … when we reverse the ordered pairs, the result is Not a function We would say the function is not one-to-one A function is one-to-one when different inputs always result in different outputs xY

One-to-One Functions When different inputs produce the same output Then an inverse of the function does not exist When different inputs produce different outputs Then the function is said to be “one-to-one” Every one-to-one function has an inverse

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

To be a one-to-one 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. This is a many-to-one function For any y value, a horizontal line will only intersection the graph once so will only have one x value This is a one-to-one 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

Graphically, the x and y values of a point are switched. The point (4, 7) has an inverse point of (7, 4) AND The point (-5, 3) has an inverse point of (3, -5)

The graph of a function and its inverse are mirror images about the line y = x y = f(x) y = f -1 (x) y = x

Steps for Finding the Inverse of a One-to-One Function Replace f(x) with y Trade x and y places Solve for yy = f -1 (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

Find the inverse of a function : y = 6x - 12 y = 6x - 12 Step 1: Switch x and y: x = 6y - 12 Step 2: Solve for y:

Given the function : y = 3x find the inverse: Step 1: Switch x and y: x = 3y Step 2: Solve for y:

Ex: (a)Find the inverse of f(x)=x y = x 5 2. x = y 5 3. (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.

Composition of Inverse Functions Consider f(3) = 27 and f -1 (27) = 3 Thus, f(f -1 (27)) = 27 and f -1 (f(3)) = 3 In general f(f -1 (n)) = n and f -1 (f(n)) = n (assuming both f and f -1 are defined for n)

Ex: Verify that f(x)=-3x+6 and g(x)= -1 / 3 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(- 1 / 3 x+2)+6 = x-6+6 = x g(f(x))= - 1 / 3 (-3x+6)+2 = x-2+2 = x ** Because f(g(x))=x and g(f(x))=x, they are inverses.

Domain and Range The domain of f is the range of f -1 The range of f is the domain of f -1 Thus... we may be required to restrict the domain of f so that f -1 is a function

Find the inverse of Steps for finding an inverse. 1.solve for x 2.exchange x’s and y’s 3.replace y with f -1 Domain of f(x) Range of f(x) Domain of f -1 (x) = Range of f(x) and Range of f -1 (x) = Domain of f(x)