Question #269503

2. The functionsf and g are defined as below. f(x) = 3x+2: XER g(x)= 6 2x +3 Find the value of x for which f(g(x)) = 3 Sketch in a single diagram, the graphs of f(x) and f(x). Express each of f(x) and g(x), and solve the equation f¹(x) = g(x)


1
Expert's answer
2021-11-22T15:55:18-0500
f(x)=3x+2,xRf(x)=3x+2, x\in \R

g(x)=62x+3,xR,x1.5g(x)=\dfrac{6}{2x+3}, x\in \R, x\not=-1.5

(i)

f(g(x))=3(62x+3)+2f(g(x)) =3(\dfrac{6}{2x+3})+2

Given

f(g(x))=3f(g(x)) =3

3(62x+3)+2=33(\dfrac{6}{2x+3})+2=3

182x+3=1\dfrac{18}{2x+3}=1

2x+3=182x+3=18

2x=152x=15

x=7.5x=7.5

(ii)


f(x)=3x+2f(x)=3x+2

Replace f(x)f(x) with yy


y=3x+2y=3x+2

Interchange xx and yy


x=3y+2x=3y+2

Solve for yy


y=13x23y=\dfrac{1}{3}x-\dfrac{2}{3}

Replace yy with f1(x)f^{-1}(x)


f1(x)=13x23f^{-1}(x)=\dfrac{1}{3}x-\dfrac{2}{3}

The graph of a function and its inverse are symmetric with respect to the line y=x.y=x.




(iii)


g(x)=62x+3,xR,x1.5g(x)=\dfrac{6}{2x+3}, x\in \R, x\not=-1.5

Replace g(x)g(x) with yy


y=62x+3y=\dfrac{6}{2x+3}

Interchange xx and yy


x=62y+3x=\dfrac{6}{2y+3}

Solve for yy


2y+3=6x2y+3=\dfrac{6}{x}

2y=6x32y=\dfrac{6}{x}-3


y=3x1.5y=\dfrac{3}{x}-1.5

Replace yy with g1(x)g^{-1}(x)


g1(x)=3x1.5,x0g^{-1}(x)=\dfrac{3}{x}-1.5, x\not=0


Given


f1(x)=g1(x)f^{-1}(x)=g^{-1}(x)

13x23=3x1.5\dfrac{1}{3}x-\dfrac{2}{3}=\dfrac{3}{x}-1.5

x22x=94.5xx^2-2x=9-4.5x

x2+2.5x9=0x^2+2.5x-9=0

D=(2.5)24(1)(9)=42.25D=(2.5)^2-4(1)(-9)=42.25

x=2.5±42.252(1)=1.25±3.25x=\dfrac{-2.5\pm\sqrt{42.25}}{2(1)}=-1.25\pm3.25

x1=1.253.25=4.5,x2=1.25+3.25=2x_1=-1.25-3.25=-4.5, x_2=-1.25+3.25=2

x{4.5,2}x\in\{-4.5, 2\}


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