Components of Composite Functions (3.18.3)

Question 1

• The benefit of composing functions with each other is that you can follow one input through several processes to reach a final output.

Answer 1

• The benefit of composing functions with each other is that you can follow one input through several processes to reach a final output.

Question 2

• Composition notation: indicates that f (x) will be evaluated with the output from g (x).

Answer 2

• Composition notation: indicates that f (x) will be evaluated with the output from g (x).

Question 3

• Note carefully which function is going to be used first. Make sure that function’s output is used in the second function.

Answer 3

• Note carefully which function is going to be used first. Make sure that function’s output is used in the second function.

Question 4

You can even create a composition of a function with itself. In this example, you are asked for at x = 2.
First evaluate f(x) at x = 2. The answer is 5. Then evaluate f (x) at x = 5. The answer is 14. 14 is the value of . In this case, you are to evaluate when x = 2. Evaluate g (x) at x = 2. The answer is 11. Then, evaluate g (x) again at x = 11. The answer is 254. 254 is the value of when x = 2.
In this example, you are to evaluate when x = x.
Following your process, evaluate g (x) when x = x. It is given that g (x)= 2x^2 + x + 1. Now, substitute all of that into f (x) in place of every x you see. Do any multiplying and combine like terms. You get 6x^2 + 3x + 2. Therefore, the value of (x)
when x = x is 6x^2 + 3x + 2.

Now, let’s reverse the composition. This time, evaluate
when x = x.

Starting with f (x) when x = x, there is no change. Substitute the entire f (x) expression for every x in g (x) and evaluate. The answer is very different from the answer you found for the composition .

Answer 4

You can even create a composition of a function with itself. In this example, you are asked for at x = 2.
First evaluate f(x) at x = 2. The answer is 5. Then evaluate f (x) at x = 5. The answer is 14. 14 is the value of . In this case, you are to evaluate when x = 2. Evaluate g (x) at x = 2. The answer is 11. Then, evaluate g (x) again at x = 11. The answer is 254. 254 is the value of when x = 2.
In this example, you are to evaluate when x = x.
Following your process, evaluate g (x) when x = x. It is given that g (x)= 2x^2 + x + 1. Now, substitute all of that into f (x) in place of every x you see. Do any multiplying and combine like terms. You get 6x^2 + 3x + 2. Therefore, the value of (x)
when x = x is 6x^2 + 3x + 2.

Now, let’s reverse the composition. This time, evaluate
when x = x.

Starting with f (x) when x = x, there is no change. Substitute the entire f (x) expression for every x in g (x) and evaluate. The answer is very different from the answer you found for the composition .