Back{Use of Tech} Fixed points of quadratics and quartics Let f(x) = ax(1 -x), where a is a real number and 0 ≤ a ≤ 1. Recall that the fixed point of a function is a value of x such that f(x) = x (Exercises 48–51).
b. Consider the polynomial g(x) = f(f(x)). Write g in terms of a and powers of x. What is its degree?
Composition of Functions
In Exercises 39 and 40, find
a. (ƒ ○ g) (-1).
ƒ(x) = 1/x , g(x) = 1/√ x + 2
Composition of Functions
In Exercises 39 and 40, find
d. (g ○ g) (x).
ƒ(x) = 1/x , g(x) = 1/√ x + 2
In Exercises 41 and 42, (a) write formulas for ƒ ○ g and g ○ ƒ and find the (b) domain and (c) range of each.
ƒ(x) = 2 - x², g(x) = √ x + 2
Graph ƒ₁ and ƒ₂ together. Then describe how applying the absolute value function in ƒ₂ affects the graph of ƒ₁.
ƒ₁(x) ƒ₂(x)
x² |x|²
Graph ƒ₁ and ƒ₂ together. Then describe how applying the absolute value function in ƒ₂ affects the graph of ƒ₁.
ƒ₁(x) ƒ₂(x)
x³ |x³|
Graph ƒ₁ and ƒ₂ together. Then describe how applying the absolute value function in ƒ₂ affects the graph of ƒ₁.
ƒ₁(x) ƒ₂(x)
4 - x² |4 - x²|
Graph ƒ₁ and ƒ₂ together. Then describe how applying the absolute value function in ƒ₂ affects the graph of ƒ₁.
ƒ₁(x) ƒ₂(x)
__ ___
√ x √ |x|
Algebraic Combinations
In Exercises 1 and 2, find the domains of f, g, f + g, and f ⋅ g.
f(x) = √(x + 1), g(x) = √(x − 1)
Algebraic Combinations
In Exercises 3 and 4, find the domains of f, g, f/g and g/f.
f(x) = 1, g(x) = 1 + √x
Composition of Functions
Let f(x) = x − 3, g(x) = √x, h(x) = x³, and j(x) = 2x. Express each of the functions in Exercises 11 and 12 as a composition involving one or more of f, g, h, and j.
c. y = x¹/⁴
Composition of Functions
Let f(x) = x − 3, g(x) = √x, h(x) = x³, and j(x) = 2x. Express each of the functions in Exercises 11 and 12 as a composition involving one or more of f, g, h, and j.
a. y = 2x − 3
Composition of Functions
Let f(x) = x − 3, g(x) = √x, h(x) = x³, and j(x) = 2x. Express each of the functions in Exercises 11 and 12 as a composition involving one or more of f, g, h, and j.
f. y = √(x³ − 3)
Composition of Functions
Copy and complete the following table.
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Composition of Functions
Copy and complete the following table.
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