Question

question 12 of 15
the functions s and t are defined as follows.
$s(x)=-x+2$
$t(x)=-2x^2 + 1$
find the value of $s(t(4))$.

question 13 of 15
suppose $h(x)=(7x-8)^5$.
find two functions $f$ and $g$ such that $(f\circ g)(x)=h(x)$.
neither function can be the identity function.
(there may be more than one correct answer.)

question 14 of 15
suppose that the functions q and r are defined as follows.
$q(x)=-5x+4$
$r(x)=2x-2$
find the following.
$(q\circ r)(-5)$
$(r\circ q)(-5)$

question 15 of 15
suppose that the functions u and w are defined as follows.
$u(x)=-2x+2$
$w(x)=x^2$
find the following.
$(w\circ u)(4)$
$(u\circ w)(4)$

Explanation:

(Question 12):

Step1: Calculate t(4)

$t(4) = -2(4)^2 + 1 = -2(16) + 1 = -32 + 1 = -31$

Step2: Calculate s(t(4))

$s(-31) = -(-31) + 2 = 31 + 2 = 33$

(Question 13):

Step1: Identify inner function g(x)

Choose $g(x) = 7x - 8$

Step2: Identify outer function f(x)

Choose $f(x) = x^5$
Verify: $(f \circ g)(x) = f(g(x)) = (7x - 8)^5 = H(x)$

(Question 14):

Step1: Calculate $(q \circ r)(-5)$

First find $r(-5) = 2(-5) - 2 = -10 - 2 = -12$
Then $q(-12) = -5(-12) + 4 = 60 + 4 = 64$

Step2: Calculate $(r \circ q)(-5)$

First find $q(-5) = -5(-5) + 4 = 25 + 4 = 29$
Then $r(29) = 2(29) - 2 = 58 - 2 = 56$

(Question 15):

Step1: Calculate $(w \circ u)(4)$

First find $u(4) = -2(4) + 2 = -8 + 2 = -6$
Then $w(-6) = (-6)^2 = 36$

Step2: Calculate $(u \circ w)(4)$

First find $w(4) = (4)^2 = 16$
Then $u(16) = -2(16) + 2 = -32 + 2 = -30$

Answer:

Question 12: $33$
Question 13: $f(x)=x^5$, $g(x)=7x-8$ (one valid pair)
Question 14: $(q \circ r)(-5)=64$, $(r \circ q)(-5)=56$
Question 15: $(w \circ u)(4)=36$, $(u \circ w)(4)=-30$