Composite Functions

Question 1

Let f(x) = 2x and g(x) = x^2. How will the equation of f(g(x)) look?

Answer 1

f(g(x)) = f(x^2) = 2(x^2).

Question 2

What restriction do rational functions have?

Answer 2

Division by 0.

Question 3

Let f(x) = 2x and g(x) = x^2, how would the expression on the right side of (f+g)(x) look?

Answer 3

(f+g)(x) = 2x + x^2

Question 4

What’s the notation for subtracting two functions. Let f and g be these two functions.

Answer 4

(f-g)(x).

Question 5

Let f(x) = 2x, g(x) = x^2, and y = (f○g)(x). What will be the value of y if x = 1?

Answer 5

(f○g)(1) = f(g(1)) = f(1) = 2.

Question 6

Let f(x) = 2x and g(x) = x^2, how would the expression on the right side of (f/g)(x) look?

Answer 6

(f/g)(x) = 2x/x^2 = 2/x

Question 7

What’s the notation for dividing two functions. Let f and g be these two functions.

Answer 7

(f/g)(x).

Question 8

Let f(x) = 3/x-3 and g(x) = 1/x. What’s the domain of f(g(x))?

Answer 8

g(x) cannot be equal to:
0, 1/3
f(x) cannot be equal to:
g(x) = 3
domain (-∞, 0) U (0, 1/3) U (1/3, ∞)

Question 9

True or false: (f○g)(x) = (g○f)(x).

Answer 9

False.

Question 10

What’s the notation for adding two functions. Let f and g be these two functions.

Answer 10

(f+g)(x).

Question 11

What’s a function with an unrestricted domain?

Answer 11

A function that accepts any input that doesn’t violate its inherent rules (division by zero or taking the square root of a negative number).

Question 12

Let f(x) = 2x, g(x) = x^2, and y = f(g(x)). What’s the value of y if x = 2?

Answer 12

f(g(x)) = f(4) = 8.

Question 13

What’s a function with a restricted domain? Provide an example.

Answer 13

A function with a restricted domain has specific input values it can take.
E(x)=−2.6x+49x+3994 models the births of Americans, where
𝑥 is the number of years after 2000. The domain is [0,11], representing years from 2000 to 2011.

Question 14

Let f and g be functions. How is f(g(x)) read?

Answer 14

f of g of x.

Question 15

What restrictions do radical functions have?

Answer 15

Taking the square root of a negative number.

Question 16

Let f and g be functions. How is the domain of the composite function f(g(x)) determined?

Answer 16

1. If x is not in the domain of g it must be excluded from the domain of f(g(x)).
2. If g(x) is not in the domain of f it must be excluded from the domain of f(g(x)).

Question 17

What’s the domain of two functions combined together via addition, or subtraction, or multiplicaiton, or division? Let these two functions be f and g.

Answer 17

The domain is all the numbers common to both f and g, i.e their intersection is the domain.

Question 18

What’s a composite function?

Answer 18

A function created by two or more functions. It is also called a nested function.

Question 19

Let f(x) = 2x and g(x) = x^2, how would the expression on the right side of (f-g)(x) look?

Answer 19

(f-g)(x) = 2x - x^2

Question 20

Let f and g be functions. If g is nested in f, how would that look in the official notation for composite functions?

Answer 20

(f○g)(x).

Question 21

What’s the notation for multiplying two functions. Let f and g be these two functions.

Answer 21

(f*g)(x).

Question 22

Let f(x) = 2x and g(x) = x^2, how would the expression on the right side of (f*g)(x) look?

Answer 22

(f*g)(x) = 2x * x^2 = 2x^3

Question 23

What’s the domain of a function?

Answer 23

All input values of a function.

Question 24

Let f and g be functions. How would a composite function of these two functions look like?

Answer 24

f(g(x)).