Suppose that the functions q and r are defined as follows:
q(x)=2x+1
r(x)=-2x^2+1
Find the following:
(qOr)(4)=
(rOq)(4)=
what is O?
I figured it out.
To find (qOr)(4), we need to find the composition of the functions q and r, evaluated at x = 4.
First, let's find (qOr)(x):
(qOr)(x) = q(r(x))
Now substitute the function r(x) into q:
(qOr)(x) = q(-2x^2 + 1)
Now substitute x = 4 into (qOr)(x):
(qOr)(4) = q(-2(4)^2 + 1)
Simplifying further:
(qOr)(4) = q(-2(16) + 1)
(qOr)(4) = q(-32 + 1)
(qOr)(4) = q(-31)
Substitute the definition of q(x):
(qOr)(4) = 2(-31) + 1
(qOr)(4) = -62 + 1
(qOr)(4) = -61
Therefore, (qOr)(4) = -61.
To find (rOq)(4), we need to find the composition of the functions r and q, evaluated at x = 4.
First, let's find (rOq)(x):
(rOq)(x) = r(q(x))
Now substitute the function q(x) into r:
(rOq)(x) = r(2x + 1)
Now substitute x = 4 into (rOq)(x):
(rOq)(4) = r(2(4) + 1)
Simplifying further:
(rOq)(4) = r(8 + 1)
(rOq)(4) = r(9)
Substitute the definition of r(x):
(rOq)(4) = -2(9)^2 + 1
(rOq)(4) = -2(81) + 1
(rOq)(4) = -162 + 1
(rOq)(4) = -161
Therefore, (rOq)(4) = -161.
To find the value of (qOr)(4), we need to first evaluate q(4) and then substitute the result into the function r(x). Similarly, to find the value of (rOq)(4), we evaluate r(4) and then substitute the result into the function q(x).
1. Finding (qOr)(4):
Step 1: Evaluate q(4):
q(x) = 2x + 1
q(4) = 2(4) + 1
q(4) = 8 + 1
q(4) = 9
Step 2: Substitute q(4) into r(x):
r(x) = -2x^2 + 1
r(q(4)) = -2(9)^2 + 1
r(q(4)) = -2(81) + 1
r(q(4)) = -162 + 1
r(q(4)) = -161
Therefore, (qOr)(4) = -161.
2. Finding (rOq)(4):
Step 1: Evaluate r(4):
r(x) = -2x^2 + 1
r(4) = -2(4)^2 + 1
r(4) = -2(16) + 1
r(4) = -32 + 1
r(4) = -31
Step 2: Substitute r(4) into q(x):
q(x) = 2x + 1
q(r(4)) = 2(-31) + 1
q(r(4)) = -62 + 1
q(r(4)) = -61
Therefore, (rOq)(4) = -61.