Question

question 23
given that \\( f\left(x\
ight)=x^{2}-9x \\) and \\( g\left(x\
ight)=x + 1 \\), calculate
(a) \\( (f\circ g)(3)= \\)
(b) \\( (g\circ f)(3)= \\)
question help: video
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question 24
given that \\( f\left(x\
ight)=x^{2}+8x \\) and \\( g\left(x\
ight)=x + 1 \\), perform the indicated operation.
(a) \\( (f\circ g)(x)= \\)
(b) \\( (g\circ f)(x)= \\)
(c) \\( (f\circ f)(x)= \\)
(d) \\( (g\circ g)(x)= \\)
(e) \\( (f\circ g)(-10)= \\)
(f) \\( (g\circ f)(-8)= \\)
question help: video 1 video 2

Explanation:

Question 23

(a) Step1: Compute $g(3)$

$g(3)=3+1=4$

(a) Step2: Compute $f(g(3))$

$f(4)=4^2-9(4)=16-36=-20$

(b) Step1: Compute $f(3)$

$f(3)=3^2-9(3)=9-27=-18$

(b) Step2: Compute $g(f(3))$

$g(-18)=-18+1=-17$

Question 24

(a) Step1: Substitute $g(x)$ into $f$

$(f\circ g)(x)=f(x+1)=(x+1)^2+8(x+1)$

(a) Step2: Expand and simplify

$(x^2+2x+1)+8x+8=x^2+10x+9$

(b) Step1: Substitute $f(x)$ into $g$

$(g\circ f)(x)=g(x^2+8x)=(x^2+8x)+1$

(b) Step2: Simplify

$x^2+8x+1$

(c) Step1: Substitute $f(x)$ into $f$

$(f\circ f)(x)=f(x^2+8x)=(x^2+8x)^2+8(x^2+8x)$

(c) Step2: Expand and simplify

$(x^4+16x^3+64x^2)+8x^2+64x=x^4+16x^3+72x^2+64x$

(d) Step1: Substitute $g(x)$ into $g$

$(g\circ g)(x)=g(x+1)=(x+1)+1$

(d) Step2: Simplify

$x+2$

(e) Step1: Use $(f\circ g)(x)$ from (a)

$(f\circ g)(-10)=(-10)^2+10(-10)+9$

(e) Step2: Calculate the value

$100-100+9=9$

(f) Step1: Use $(g\circ f)(x)$ from (b)

$(g\circ f)(-8)=(-8)^2+8(-8)+1$

(f) Step2: Calculate the value

$64-64+1=1$

Answer:

Question 23:
(a) $-20$
(b) $-17$

Question 24:
(a) $x^2+10x+9$
(b) $x^2+8x+1$
(c) $x^4+16x^3+72x^2+64x$
(d) $x+2$
(e) $9$
(f) $1$