Question

Part A

Consider functions m and n: n(x)=1/4x^2-2x+4

The value of m(n(2)) is __

The value of n(m(1)) is __


Part B

Consider the functions m and n

n(x)=1/4x^2-2x+4

What is the value of n(m(4))

A. -4

B. -2

C. 0

D. 4


Part C

Given your answers to Katy’s A and B, do you think functions m and n are inverse functions? Explain your reasoning

Answer 1

Answer: To be inverse functions, the values of n(m(x)) and m(n(x)) must equal x for all values of x in the domain. In part A, there were two cases where the value of the composite function was equal to x. In part B, however, there was a case where n(m(x)) was not equal to x. Therefore, the two functions cannot be inverse functions.

Step-by-step explanation: answer to question C

Answer 2

Part A:

To find the value of m(n(2)), we need to first find the value of n(2) and then use that value to find m.

n(2) = 1/4(2)^2 - 2(2) + 4

= 1/4(4) - 4 + 4

= 1 - 4 + 4

= 1

So, n(2) = 1.

Now, we can find m(1) using the equation for m:

m(1) = 3 - 2(1) + 4

= 5

Therefore, m(n(2)) = m(1) = 5.

To find the value of n(m(1)), we need to first find the value of m(1) and then use that value to find n.

m(1) = 3 - 2(1) + 4

= 5

So, m(1) = 5.

Now, we can find n(5) using the equation for n:

n(5) = 1/4(5)^2 - 2(5) + 4

= 1/4(25) - 10 + 4

= 6.25 - 10 + 4

= 0.25

Therefore, n(m(1)) = n(5) = 0.25.

Part B:

To find the value of n(m(4)), we need to first find the value of m(4) and then use that value to find n.

m(4) = 3 - 2(4) + 4

= -1

So, m(4) = -1.

Now, we can find n(-1) using the equation for n:

n(-1) = 1/4(-1)^2 - 2(-1) + 4

= 1/4(1) + 2 + 4

= 1.25 + 2 + 4

= 7.25

Therefore, n(m(4)) = n(-1) = 7.25.

The answer is not one of the options provided.

Part C:

The functions m and n are inverse functions if and only if applying them in either order gives the identity function, i.e., m(n(x)) = x and n(m(x)) = x for all x in the domain of the functions.

From our calculations in Part A, we know that m(n(2)) = 5 and n(m(1)) = 0.25, which means that m(n(x)) ≠ x and n(m(x)) ≠ x for some values of x in the domain of the functions. Therefore, we can conclude that functions m and n are not inverse functions.