Combining functions algebraically, composite functions, and decomposing functions! Onward to Section 1.4a…

Definition: Sum, Difference, Product, and Quotient of Functions Let f and g be two functions with intersecting domains. Then for all values of x in the intersection, the algebraic combinations of f and g are defined by the following rules: Sum: In each case, the domain of the new function consists of all numbers that belong to both the domain of f and the domain of g. As noted, the zeros of the denominator are excluded from the domain of the quotient. Difference: Product: Quotient: provided

Guided Practice For the given functions, find f + g, f – g, fg, f/g, and gg. Give the domain of each. First, find the domain of the original functions, and determine where these two domains intersect (overlap). Domain of f:Domain of g: Domain intersection: This intersection becomes the domain of all of the algebraic combination functions!!!

Guided Practice For the given functions, find f + g, f – g, fg, f/g, and gg. Give the domain of each. with D:

Guided Practice For the given functions, find f + g, f – g, fg, f/g, and gg. Give the domain of each. with D: Can we simplify this last one???

Guided Practice For the given functions, find formulas for the functions f + g, f – g, and fg. Give the domain of all functions. Domain of all 3 combination functions:

Guided Practice For the given functions, find formulas for f/g and g/f. Give the domain of all functions.

Now on to composite functions?!

Definition: Composition of Functions Let f and g be two functions such that the domain of f intersects the range of g. The composition of f and g, denoted f g, is defined by the rule The domain of f g consists of all x-values in the domain of g that map to g(x)-values in the domain of f. NOTE: In most cases, f g and g f are different functions!!!

A Few Practice Problems… For the given functions, find (f g)(x) and (g f)(x) and verify (both algebraically and graphically) that the two composite functions are not the same. Now, how do we verify???

A Few Practice Problems… For the given functions, find (f g)(x) and (g f)(x) and give the domain of each composition function. D: Let’s check these with the calculator…

A Few Practice Problems… For the given functions, find (f g)(3) and (g f)(–2).

Decomposing Functions…working backwards or undoing a composition… For each function h, find functions f and g such that h(x) = f(g(x)).

Decomposing Functions For each function h, find functions f and g such that h(x) = f(g(x)). Any other ways to solve this one?!?!

Let’s do some modeling… In the medical procedure known as angioplasty, doctors insert a catheter into a heart vein and inflate a small, spherical balloon on the tip of the catheter. Suppose the balloon is inflated at a constant rate of 44 cubic millimeters per second. In math-land, not fashion-land… 1. Find the volume after t seconds. V = 44t

Let’s do some modeling… In the medical procedure known as angioplasty, doctors insert a catheter into a heart vein and inflate a small, spherical balloon on the tip of the catheter. Suppose the balloon is inflated at a constant rate of 44 cubic millimeters per second. In math-land, not fashion-land… 2. When the volume is V, what is the radius r ?

Let’s do some modeling… In the medical procedure known as angioplasty, doctors insert a catheter into a heart vein and inflate a small, spherical balloon on the tip of the catheter. Suppose the balloon is inflated at a constant rate of 44 cubic millimeters per second. In math-land, not fashion-land… 3. Write an equation that gives the radius r as a function of the time. What is the radius after 5 seconds? and At 5 seconds, r = mm

Whiteboard Practice For the given functions, find formulas for the functions f + g, f – g, and fg. Give the domain of all functions. Domain of all five functions:

Find f(g(x)) and g(f(x)). State the domain of each. Whiteboard Practice

Find f(x) and g(x) so that the function can be described as y=f(g(x)). one possible solution… Homework: p odd Whiteboard Practice