2.7 Composition of functions
Define function composition.
Consider the composite function h(x)=(x-3)^2+(x-3)+1, where h(x)=f(g(x)).
State the input of f(x).
State the input of g(x).
Given the following functions, determine the expression that defines each function composition.
- f(x)=3x^2+5x+4
- g(x)=-2\sqrt{x}+1
- h(x)=7x^2
(f\circ g)(x)
(g\circ f)(x)
(f\circ f)(x)
\left(f(g(h))\right)(x)
Consider the functions A\left(x\right) = 4x^2 + 3 and B\left(x\right) = x - 5.
State the domain of A.
State the domain of B.
Find an expression for \left(A (B)\right)\left(x\right).
State the domain of the function \left(B(A)\right).
The table shows some of the outputs of the functions f, g, and h.
Use the table to evaluate the following:
| x | f\left(x\right) | g\left(x\right) | h\left(x\right) |
|---|---|---|---|
| 0 | 1 | 8 | 2 |
| 1 | -1 | 8 | 4 |
| 2 | 0 | 5 | 8 |
| 4 | 2 | 0 | 11 |
| 8 | 16 | -2 | 5 |
| 16 | 64 | -12 | 2 |
Given the functions: f(x) = x^2 + 2x + 1 and g(x) = x - 1.
Find (f \circ g)(x).
Find (g \circ f)(x).
Explain why these composite functions are not equivalent.
Using the given set of functions, evaluate the following composite functions.
(f \circ g)(x)
(f \circ f)(x)
(f\circ g\circ h)(x)
(h\circ g\circ f)(x)
f(x) = 3x^2 - 2x + 1, g(x) = -x+4, and h(x) = 5x
f(x) = 9x + 7, g(x) = x^2 - 6 and h(x)=x+4
f(x) = \sqrt x +3, g(x) = 16x^2 , and h(x) = 3x - 1
f(x) = 4x+5, g(x) = x^2 - 2 and h(x) = \sqrt x
Using the given set of functions, evaluate the following composite functions.
(f \circ g)(2)
(h \circ f)(2)
(f\circ g\circ h)(2)
(h\circ g\circ f)(2)
f(x) = x^3 + x^2 + x + 1, g(x) = - 2x + 1, and h(x) = - 4x
f(x) = - 4x + 4, g(x) = 2x^2 - x + 3, and h(x) = x^3
f(x) = x^2 - 3, g(x) = \sqrt x, and h(x) = 2x + 6
f(x) = \sqrt x, g(x) = x^2 + 1, and h(x) = x - 9
Given the function f(x) = x^2.
Describe the transformation represented by the composition of g(x) = x + 3 with f.
How does this transformation affect the graph of f?
Consider the functions f(x) = x^2 - 2x + 1 and g(x) = 3x + 1, determine the domain of the composite function (f \circ g)(x).
Given the function f(x) = \sqrt{x}.
Describe the transformation represented by the composition of g(x) = 2x with f.
How does this transformation affect the graph of f?
Given the function f(x) = 5x^2 + 3x - 2, show an analytic representation of the composite function f(f(x)).
The functions h(x) = 3x + 2 and j(x) = x - 4 are composed to form a new function k(x). If k(5) = 7, determine whether h was composed with j, or j was composed with h.
The functions f and g are given by the graphs below.
Based on the graphs, evaluate:
(f\circ g)(2)
(g\circ f)(-3)
(f\circ g)(1)
(g\circ f)\left(-\dfrac32\right)
Given the following composite functions h(x), decompose this into two functions f(x) and g(x) such that h(x) = f(g(x)).
h(x) = (2x^3 - x + 1)^5
h(x) = \sqrt{4x - 7}
h(x) = (3x^2 - 4x + 2)^4+8
h(x) =2\sqrt{x^4 - 2x^2 + x - 1}
Explain why function composition is not commutative. Use an example to illustrate your answer.
Consider the function h(x) = 4x^{2} - 2x + 1.
When composing the function h(x) with the identity function g(x) = x, what is the resulting function?
How does this demonstrate the role of the identity function in function composition?
The distance, D, in meters that a car travels is given by D = v \times t where v is the velocity in \text{m/s} and t is the time in seconds. The velocity of the car changes with time according to the function v = a \times t, where a is the acceleration in \text{m/s}^2.
Write a composite function that gives the distance travelled by the car in terms of acceleration and time.
Find the distance if the acceleration of the car is 2 \text{ m/s}^2 in 5 seconds.
A coffee shop sells coffee at \$2 per cup and pastries at \$3 each. The function f(x) denotes the total cost of x cups of coffee and g(x) denotes the total cost of x pastries. If a customer orders 3 cups of coffee and 2 pastries, calculate the total cost as a composition of the two functions, f(g(x)).
A gym charges a monthly membership fee of \$30 and an additional \$10 fee per class attended. Let's denote the monthly membership fee function as f(x) and the per class fee as g(x). If a member attends 5 classes in a month, what is the total cost? Calculate this as a composition of two functions, f(g(x)).