Section 1-5 Combinations of Functions PreCalculus Section 1-5 Combinations of Functions

Objectives Add, subtract, multiply, and divide functions. Find compositions of one function with another function. The combinations of functions to model and solve real-life problems.

Combinations of Functions Just as two real numbers can be combined by the operations of addition, subtraction, multiplication, and division to form other real numbers, two functions can be combined by those same operations to create new functions. In addition, functions can be combined using the function operation “composition of functions.”

Sums, Differences, Products, and Quotients of Functions If f and g are functions and x is in the domain of each function, then Domain for sums, differences, products and quotients of functions is the intersection of the domains of functions f and g (Df∩Dg).

The Sum of Two Functions

The sum f + g Combine like terms & put in descending order This just says that to find the sum of two functions, add them together. You should simplify by finding like terms. Combine like terms & put in descending order

Example:

Your Turn: Given f(x)=x2 and g(x)=1-x, find (f+g)(x), (f+g)(4) and the domain of (f+g). Solution: (f+g)(x)=x2-x+1 (f+g)(4)=13 D(f+g)=all real numbers

Example: Addition of Functions on a Graph Use the graphs of f and g to graph h(x) = (f + g)(x). Measure the distance from the x-axis to the first point on g. Add this distance to the first point on f. Do this for each additional point. h Connect the dots and you have h(x) = f(x) = g(x).

Your Turn: Use the graphs of f and g to graph h(x) = (f + g)(x). h

The Difference of Two Functions

The difference f - g Distribute negative To find the difference between two functions, subtract the second from the first. CAUTION: Make sure you distribute the – to each term of the second function. You should simplify by combining like terms. Distribute negative

Example Given that f(x) = x2 + 2 and g(x) = x  3, find each of the following. a) (f  g)(x) b) The domain of f  g Solution: a) (f  g)(x) = f(x)  g(x) = (x2 + 2)  (x  3) = x2  x + 5 b) The domain of f is the set of all real numbers. The domain of g is also the set of all real numbers. The domain of f  g is the set of numbers in the intersection of the domains—that is, the set of numbers in both domains, or all real numbers.

Your Turn: Given f(x)=x2 and g(x)=1-x, find (f-g)(x) and D(f-g). Solution: (f-g)(x)=x2+x-1 D(f-g)={x|x∈ℝ}

The Product of Two Functions

The product f • g Multiply To find the product of two functions, put parenthesis around them and multiply each term from the first function to each term of the second function. Multiply Good idea to put in descending order but not required.

Example: ∙ ∙

Your Turn: Find (gf)(x) and its domain given that f(x)=1/x and g(x)=x/(x+1). Solution: (gf)=1/(x+1) D(gf)={x|x≠0, x≠-1}

The Quotient of Two Functions

The quotient f /g To find the quotient of two functions, put the first one over the second. Nothing more you could do here. (If you can reduce these you should).

Example: Solution (a) (b)

Your Turn: Given f(x)=x2 and g(x)=1-x, find f/g(x) and the domain of f/g. Solution:

Example Given that f(x) = x + 2 and g(x) = 2x + 5, find each of the following. a) (f + g)(x) b) (f + g)(5) Solution: a) b) We can find (f + g)(5) provided 5 is in the domain of each function. This is true. f(5) = 5 + 2 = 7 g(5) = 2(5) + 5 = 15 (f + g)(5) = f(5) + g(5) = 7 + 15 = 22 or (f + g)(5) = 3(5) + 7 = 22

Example: Evaluating Combinations of Functions on a Graph If possible, use the given representations of functions f and g to evaluate;

Your Turn: If possible, use the given representations of functions f and g to evaluate; = -1 DNE = 3 = -2

Finding the composition of two functions. Composite functions Finding the composition of two functions.

Composition of Functions Definition:

What is a Composite Function? COMPOSITE FUNCTION - two functions put together. One function (f) is the input to another function (g). gof means f then g. Alternately written just as g(f(x)).

Composition of Functions Composition of functions is the successive application of the functions in a specific order. Given two functions f and g, the composite function is defined by and is read “f of g of x.” The domain of is the set of elements x in the domain of g such that g(x) is in the domain of f. Another way to say that is to say that “the range of function g must be in the domain of function f.”

A composite function domain of g range of f range of g domain of f x g g(x) domain of g range of g f range of f f(g(x)) domain of f

A different way to look at it… Function Machine x Function Machine g f

Evaluate Composite Function f(x)=x+2 g(x)=5x Calculate fog(2) 2 g(2) =5 x 2 10 10 f(10)=10+2 12

Your Turn: Given find (a) and (b) Solution (a) (b)

Evaluate Composite Function Find f(g(x)) same as fog(x) or (fog)(x) Finally, this simplifies to:

The Composition Function This is read “f of g of x” and means x is the input to function g and then g(x) is the input to function f. Square first and then distribute the 2

You could multiply this out but since it’s to the 3rd power we won’t

This is read “f of f of x” and means to copy the f function down but where ever you see an x, substitute in the f function. (So sub the function into itself).

Example Given that f(x) = 3x  1 and g(x) = x2 + x  3, find: a) b)

Example: Finding Composite Functions Let and Find (a) and (b) Solution (a) (b) Note:

Your Turn:

The DOMAIN of the Composition Function The domain of f ∘g is the set of all numbers x in the domain of g such that g(x) is in the domain of f. The domain of g is x  1 We also have to worry about any “illegals” in this composition function, specifically dividing by 0. This would mean that x  1 so the domain of the composition would be combining the two restrictions.

The DOMAIN and RANGE of Composite Functions We could first look at the natural domain and range of f(x) and g(x). For g(x) to cope with the output from f(x) we must ensure that the output does not include 1 Hence we must exclude 6 from the domain of f(x)

The DOMAIN and RANGE of Composite Functions Or we could find g o f (x) and determine the domain and range of the resulting expression. Domain: Range: However this approach must be used with CAUTION.

The DOMAIN and RANGE of Composite Functions We could first look at the natural domain and range of f(x) and g(x). For f(x) to cope with the output from g(x) we must ensure that the output does not include 0 Hence we must exclude 1 from the domain of g(x)

The DOMAIN and RANGE of Composite Functions Or we could find f o g (x) and determine the domain and range of the resulting expression. Domain: Range: However this approach must be used with CAUTION.

The DOMAIN and RANGE of Composite Functions We could first look at the natural domain and range of f(x) and g(x).

The DOMAIN and RANGE of Composite Functions Or we could find g o f (x) and determine the domain and range of the resulting expression. Not: Domain: Range: However this approach must be used with CAUTION.

The DOMAIN and RANGE of Composite Functions We could first look at the natural domain and range of f(x) and g(x). f(x) can cope with all the numbers in the range of g(x) because the range of g(x) is contained within the domain of f(x) f o g (x) is a function for the natural domain of g(x)

The DOMAIN and RANGE of Composite Functions We could first look at the natural domain and range of f(x) and g(x). g(x) cannot cope with all the numbers in the range of f(x). Need to restrict the domain f(x) to give an output that g(x) can cope with.

The DOMAIN and RANGE of Composite Functions g o f (x) is not a function for the natural domain of g(x) unless we restrict the domain of f(x) We could first look at the natural domain and range of f(x) and g(x). g(x) cannot cope with all the numbers in the range of f(x). Need to restrict the domain f(x) to give an output that g(x) can cope with.

Example: Finding the Composite of Functions on a Graph If possible, use the given representations of functions f and g to evaluate; (f ∘g)(1) = 3 (f ∘g)(0) = 5 (g ∘f )(6) = DNE (g ∘f )(3) = 0 (f ∘g)(x)

Your Turn: (f ∘g)(3) = 4 (f ∘g)(7) = DNE (f ∘g)(4) = 1 (g ∘f )(6) = 1 If possible, use the given representations of functions f and g to evaluate; (f ∘g)(3) = 4 (f ∘g)(7) = DNE (f ∘g)(4) = 1 (g ∘f )(6) = 1 (g ∘f )(3) = 2

Decomposing a Function as a Composition In calculus, one needs to recognize how a function can be expressed as the composition of two functions. Example: If h(x) = (3x  1)4, find f(x) and g(x) such that Solution: The function h(x) raises (3x  1) to the fourth power. Two functions that can be used for the composition are: f(x) = x4 and g(x) = 3x  1.

Your Turn: Suppose that Find f and g so that Solution Note the repeated quantity Let Note that there are other pairs of f and g that also work.

Application of Composition of Functions Suppose an oil well off the California coast is leaking. Leak spreads in circular layer over water Area of the circle is At any time t, in minutes, the radius increases 5 feet every minute. Radius of the circular oil slick is Express the area as a function of time using substitution.

Applying a Difference of Functions Example The surface area of a sphere S with radius r is S = 4 r2. Find S(r) that describes the surface area gained when r increases by 2 inches. Determine the amount of extra material needed to manufacture a ball of radius 22 inches as compared to a ball of radius 20 inches.

Assignment Section 1.5, pg. 58 – 61: Vocabulary Check #1 – 4 all Exercises: #1-25 odd, 35-43 odd, 61-71 odd, 77, 81 Read Section 1.6, pg. 62 – 68