8 terms

Precalculus-Combinations of Functions

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(f+g)(x) = ?

Example: Given f(x)=3x and g(x)=2x+1
f(x) + g(x)

Example: Given f(x)=3x and g(x)=2x+1
(3x)+(2x+1)
5x+1
(f-g)(x) = ?

Example: Given f(x)=3x and g(x)=2x+1
f(x) - g(x)

Example: Given f(x)=3x and g(x)=2x+1
(3x)-(2x+1)
x-1
(fg)(x) = ?

Example: Given f(x)=3x and g(x)=2x+1
f(x) · g(x)

Example: Given f(x)=3x and g(x)=2x+1
(3x)·(2x+1)
6x²+3x
(f/g)(x) = ?

Example: Given f(x)=3x and g(x)=2x+1
f(x)/g(x) , g(x) ≠ 0

Example: Given f(x)=3x and g(x)=2x+1
(3x)/(2x+1) , x≠-½
What do you have to look for when finding the domain of a quotient?

Example: What is the domain of (f/g)(x) , given f(x)=√x and g(x)=√(4-x²)
1. The denominator ≠ 0
2. If there are square roots, or even roots, you cannot take the root of a negative number.

Example: (f/g)(x)=f(x)/g(x)
=( √x)/[√(4-x²)]
Our denominator ≠0 nor can we take the root of a negative number, so -2<x<2
AND
For the numerator, we cannot take the root of a negative number, so x≥0

So the domain of (f/g)(x) is 0≤x<2 OR [0,2) in interval notation
(f ∘ g)(x) = ?

Example: f(x)=x+2 and g(x)=4-x²
f(g(x))

Example: f(x)=x+2 and g(x)=4-x²
(f ∘ g)(x) = f(g(x))
= f(4-x²) Plug in g(x)
= (4-x²)+2 Plug g(x) in f(x) function
= -x²+6 Simplify
What is the domain of a composite function (f ∘ g)(x)?

Example: f(x)=x²-9 and g(x)=√(9-x²)
It is either equal to or a restriction of the domain of the inner function.

Example: f(x)=x²-9 and g(x)=√(9-x²)
(f ∘ g)(x) = f(g(x))
= f(√(9-x²))
= [√(9-x²)]²-9
= -x²
Even if the composite function is -x²,which looks like it has a domain of all real numbers, this is NOT the case.

Domain of f(x) is all real numbers and of g(x) is -3≤x≤3. Because of this restriction of the inner function, the actual domain of the composite function is -3≤x≤3
How do you decompose a composite function?

Example: h(x)=(3x-5)³
1. Look for the "inner" function
2. Look for the "outer" function.
3. Write both functions f(x) and g(x)

Example: h(x)=(3x-5)³
1. Inner functions is 3x-5
2. Outer function is x³
3. g(x)=3x-5 and f(x)=x³