Composition of Functions Suppose we have two money machines, both of which increase any money inserted into them. Machine A doubles our money while Machine B adds five dollars. The money that comes out is described by a(x) = 2x for Machine A and b(x) = x + 5 for Machine B, where x is the number of dollars inserted. The machines can be hooked together so that the money coming out of one machine goes into the other. There are two ways of hooking up the machines which result in the formulas shown below. The first formula is the composition of b with a, while the second formula is the composition of a with b. Which of these two compositions would you prefer? Why? AB$$: B A$

Decomposition of Functions Sometimes we reason backward to find the functions which went into a composition. This process is called decomposition, and there may be more than one way to do it. Problem. The vertex formula for the family of quadratic functions is p(x) = a(x – h) 2 + k. Decompose the formula into three simple functions. That is, find formulas for u, v, and w where p(x) = u(v(w(x))). Solution. We work from the inner parentheses outward. Let w(x) = x – h. Let v(x) = x 2 and let u(x) = ax + k. Now, let’s check our work: u(v(w(x))) = av(w(x)) + k = a(w(x)) 2 + k = a(x – h) 2 + k. Thus, we have correctly decomposed the formula for p(x).

Definition of Inverse Function We have previously studied inverse functions. For example, we defined the logarithm as the inverse function of the exponential function. We now give a more careful treatment of inverse functions. Suppose Q = f(t) is a function with the property that each value of Q determines exactly one value of t. Then f has an inverse function, If a function has an inverse, it is said to be invertible. Example. Solve sin x = 0.8 using an inverse function. A calculator (set in radians) gives x = sin -1 (0.8) 0.93.

Finding a formula for an inverse function In the graph below, we have P = f(t) = t, and we want to graphically find P Locate 25 on the P-axis Read off the value of t corresponding to P = 25.

If we continue with the example from the previous slide, we may also solve algebraically for The equation to be solved is: t = 25, and we first subtract 20 from both sides to obtain 0.4t = 5. Upon dividing by 0.4, we have t = This same algebraic procedure can be carried for a general P to obtain the formula for We must solve t = P for t. We have 0.4t = P – 20, and thus, That is,

Problem. Suppose we have a savings account which pays 4% interest compounded annually. The balance, in dollars, in the account after t years is given by B = f(t) = 500(1.04) t. The inverse function t = f -1 (B) gives the number of years for the balance to grow to $B. Find a formula for t = f -1 (B). Solution. We solve for t in terms of B in the given equation.

The Horizontal Line Test If there is a horizontal line which intersects a function’s graph in more than one point, then the function does not have an inverse. If every horizontal line intersects a function’s graph at most once, then the function has an inverse. The graph of the function q(x) = x 2, which is shown below, fails the horizontal line test. Consider y = 4. The line y = 4 intersects graph of q(x) = x 2 twice.

Evaluating an Inverse Function Graphically Let u(x) = x 3 + x + 1. The graph of u(x) is shown below, and it looks as though u passes the horizontal line test. To find u -1 (4), we can proceed graphically or we can use Maple. u -1 (4)  evalf(solve(x^3+x+1=4,x)); , I, I

The Graph, Domain, and Range of the Inverse Function. Suppose f is an invertible function. Then outputs of f are inputs of f -1. Similarly, outputs from f -1 are inputs of f. It follows that: Domain of f -1 = Range of f and Range of f -1 = Domain of f. Example. The domain of f(x) = 10 x is all real numbers while its range is all positive real numbers. Of course, we know that f -1 (x) = log x, and the domain of f -1 is all positive real numbers while its range is all real numbers. It is known that the graph of f -1 can be obtained by reflecting the graph of f across the line y = x.

A Property of Inverse Functions If y = f(x) is an invertible function and y = f -1 (x) is its inverse, then f -1 (f(x)) = x for all values of x for which f(x) is defined, f(f -1 (x)) = x for all values of x for which f -1 (x) is defined. Example. f(x) = x/(1+x) and f -1 (x) = x/(1–x). We verify that the first property listed above holds for these functions:

Restricting the domain A function which fails the horizontal line test is not invertible. For this reason, f(x) = x 2 does not have an inverse function. However, if we restrict the domain of f to the set of nonnegative x values, then the restricted graph does pass the horizontal line test. Thus, f(x) = x 2 does have an inverse on its restricted domain, If we solve y = x 2 for x, where we have that x = We note that is defined to be the nonnegative value whose square is y. Thus, f(x) = x 2, restricted to has the inverse function f -1 (x) = We note that the domain of f -1 is also Do you see why?

The graph of f(x) = x 2 restricted to and the graph of f -1 (x) =

Combinations of Functions Like numbers, functions can be combined using addition, subtraction, multiplication, and division. For addition, the sum of two functions is defined by adding the values of the two functions at each value of x where both functions are defined. Subtraction, multiplication, and division of two functions are handled similarly. Example. Let f(x) = x and g(x) = 1/x. Define h(x) = f(x) + g(x), for x > 0. Complete the table below for the values of h(x). x f(x) = x g(x) = 1/x h(x) = f(x)+g(x) ? 2.50 ? ? ?

Graph of h(x) = f(x) + g(x), where f(x) = x and g(x) = 1/x, x>0. x y

Factoring a Function’s Formula into a Product It is often useful to express a given function as a product of functions. Example. Find exactly all the zeros of We can factor out a common factor of 2 x : Next, we factor the quadratic: The zeros are since 2 x is never zero.

A function defined as a quotient of two functions If we let f(x) = sin x and g(x) = cos x, then The quotient function has all real numbers except odd multiples of  /2 as its domain. That is, the values where g(x) = cos x is zero must be excluded from the domain. y = tan x

The sum of two even functions is even Example.

The product of two odd functions is even Example.

Summary of Compositions,Inverses, & Combinations of Functions Two functions form a new function when the output of one becomes the input of the other--this is composition. Decomposition of functions was also introduced. Suppose Q = f(t) is a function with the property that each value of Q determines exactly one value of t. Then f has an inverse function, We discussed: Finding a formula for f -1, the horizontal line test, domain and range of f -1, and the relation of the graphs of f and f -1. The compositions of f and f -1 were formed in both possible ways. Restricting the domain of f so that the resulting function has an inverse was investigated. Combining functions using addition, subtraction, multiplication, and division was studied.