Combinations of Functions

Warm-upa) +

b) -

c) ·

Arithmetic Combinations(f+g)(x) = f(x) + g(x)(f-g)(x) = f(x) – g(x)(fg)(x) = f(x) ∙ g(x)(f/g)(x) = f(x) ; g(x) ≠0 g(x)

The domain for these combinations is all values of x common to both domains, with the additional restriction that g(x)≠0 for f/g.

Evaluate each of these combinations for f(x) = 4x – 12 and g(x) = x2 – 9.

(f+g)(x) = x2 + 4x – 21 , domain is all realscompare f(2) + g(2) to (f+g)(2)

(f-g)(x) = -x2 + 4x – 3 , domain is all reals(f·g)(x) = 4x3 -12x2 - 36x + 108 , domain is all

reals(f/g)(x) = 4/(x + 3); domain is all reals, x ≠ ±3

Look out for the domain…If f(x) = 1/(x – 5) and g(x) = √x, what is the

domain of (f+ g)(x)?

The domain of f(x) is all reals but 5. The domain of g(x) is all non-negative numbers. So the domain of the sum is [0,5) U (5, ∞).

Let f(x) = √x and g(x) = √(x + 3)

Find the domain of g/f and f/g.

The domain of f(x) is x ≥ 0. The domain of g(x) is x ≥ -3

The intersection of these is x ≥ 0.

However…

We have (f/g)(x) = √x √(x+ 3)

And in this case the domain is [0,∞).

But (g/f)(x) = √(x+ 3) √x

In this case the domain is (0, ∞). Can you see why there is a difference?

Composition of FunctionsThe composition of the function f with g is (f ° g)(x) = f(g(x)).

The domain of the function is all x that are in the domain of g such that g(x) is in the domain of f.

Let f(x) = x2 + 2x and g(x) = x + 5.

Find the f(g(1)) and g(f(1)).f(g(1)) = 48g(f(1)) = 8

Find f(g(x)) and g(f(x)).f(g(x)) = x2 + 12x + 35g(f(x)) = x2 + 2x + 5

Let f(x) = x2 – 25 and g(x) = √(25 – x2).

Simplify f(g(x)).

Find the domain of the f(g(x)).

You might think that the domain is all real numbers, but it is [-5, 5] because that is the domain of g(x).

Try graphing y = (√(25 – x2))2 – 25 and see what happens.

Let f(x) = √x and g(x) = x + 5.

What is the domain of f(g(x))?

The domain of g(x) is all real numbers. But the domain of f(x) is x ≥0. So only values of x such that g(x), which is x + 5, are greater than or equal to 0 work. So x must be greater than or equal to -5.

Or just look at the composition function (before simplifying) to find its domain.

Decomposing FunctionsWrite each function as the composition of two

or more functions.

a) f(x) = (2x+ 1)2 – 7let g(x) = 2x + 1, let h(x) = x2 – 7then h(g(x)) = (2x + 1)2 – 7(This answer is not unique.)

b) f(x) = (x – 3)2 + 2(x- 3) + 1 let g(x) = x – 3, let h(x) = x2 + 2x + 1then h(g(x)) = (x – 3)2 + 2(x- 3) + 1