Answer :
Answer:
(i) [tex]g(-2)=-2[/tex]
(ii) [tex]g[h(x)]=12x-14[/tex]
(iii) [tex]f[g(2)]=55[/tex]
(iv) [tex](g\circ h)(2)=10[/tex]
(v) [tex]h^{-1}(11)=\frac{17}{4}[/tex]
Step-by-step explanation:
The given functions are
[tex]f(x)=6x-5[/tex]
[tex]g(x)=3x+4[/tex]
[tex]h(x)=4x-6[/tex]
(i) Find g(-2).
Substitute x=-2 in g(x).
[tex]g(-2)=3(-2)+4\Rightarrow -6+4=-2[/tex]
(ii) Find g[h(x)]
[tex]g[h(x)]=g(4x-6)[/tex] [tex](h(x)=4x-6)[/tex]
[tex]g[h(x)]=3(4x-6)+4[/tex] [tex](g(x)=3x+4)[/tex]
[tex]g[h(x)]=12x-18+4[/tex]
[tex]g[h(x)]=12x-14[/tex]
(iii) Find f[g(2)].
[tex]f[g(2)]=f[3(2)+4][/tex] [tex](g(x)=3x+4)[/tex]
[tex]f[g(2)]=f(6+4)[/tex]
[tex]f[g(2)]=f(10)[/tex]
[tex]f[g(2)]=6(10)-5[/tex] [tex]f(x)=6x-5[/tex]
[tex]f[g(2)]=55[/tex]
iv) Find gᴏh(2).
[tex](g\circ h)(2)=g[h(2)][/tex]
[tex](g\circ h)(2)=12(2)-14[/tex] (From part (ii) we get [tex] g[h(x)]=12x-14)[/tex]
[tex](g\circ h)(2)=10[/tex]
(v) Find [tex]h^{-1}(11)[/tex]
First find [tex]h^{-1}(x)[/tex].
[tex]h(x)=4x-6[/tex]
[tex]y=4x-6[/tex]
[tex]x=4y-6[/tex]
[tex]x+6=4y[/tex]
Divide both sides by 4.
[tex]\frac{x+6}{4}=y[/tex]
[tex]h^{-1}(x)=\frac{x+6}{4}[/tex]
Substitute x=11 in the inverse function.
[tex]h^{-1}(11)=\frac{11+6}{4}[/tex]
[tex]h^{-1}(11)=\frac{17}{4}[/tex]