Question

suppose that the functions q and r are defined as follows.
q(x) = x² + 6
r(x) = √(x + 9)
find the following.
(q ∘ r)(7) =
(r ∘ q)(7) =

Explanation:

Step1: Define function composition

$(q \circ r)(x) = q(r(x))$, $(r \circ q)(x) = r(q(x))$

Step2: Calculate $(q \circ r)(7)$

First find $r(7)$:
$r(7) = \sqrt{7+9} = \sqrt{16} = 4$
Then substitute into $q(x)$:
$q(r(7)) = q(4) = 4^2 + 6 = 16 + 6 = 22$

Step3: Calculate $(r \circ q)(7)$

First find $q(7)$:
$q(7) = 7^2 + 6 = 49 + 6 = 55$
Then substitute into $r(x)$:
$r(q(7)) = r(55) = \sqrt{55+9} = \sqrt{64} = 8$

Answer:

$(q \circ r)(7) = 22$
$(r \circ q)(7) = 8$