QUESTION 1
i) Given [tex]f(x)=5x-2[/tex] and [tex]g(x)=x^2[/tex].
[tex](f\circ g)=f(g(x))[/tex]
[tex](f\circ g)=f(x^2)[/tex]
We plug in [tex]x^2[/tex] into [tex]f(x)=5x-2[/tex]
[tex](f\circ g)=5x^2-2[/tex]
ii) Given [tex]f(x)=5x-2[/tex] and [tex]g(x)=x^2[/tex].
[tex](g\circ f)=g(f(x))[/tex]
[tex](g\circ f)=g(5x-2)[/tex]
We plug in [tex]5x-2[/tex] into [tex]g(x)=x^2[/tex]
[tex](g\circ f)=(5x-2)^2[/tex]
[tex](g\circ f)=25x^2-10x-10x+4[/tex]
[tex](g\circ f)=25x^2-20x+4[/tex]
QUESTION 2
i) Given [tex]g(x)=x^2-4[/tex] and [tex]h(x)=\sqrt{2x+1}[/tex].
[tex](h\circ g)=h(g(x))[/tex]
[tex](h\circ g)=h(x^2-4)[/tex]
We plug in [tex]x^2-4[/tex] into [tex]h(x)=\sqrt{2x+1}[/tex]
[tex](h\circ g)=\sqrt{2(x^2-4)+1}[/tex]
[tex](h\circ g)=\sqrt{2x^2-8+1}[/tex]
[tex](h\circ g)=\sqrt{2x^2-7}[/tex]
ii) Given [tex]g(x)=x^2-4[/tex] and [tex]h(x)=\sqrt{2x+1}[/tex].
[tex](g\circ h)=g(h(x))[/tex]
[tex](g\circ h)=g(\sqrt{2x+1})[/tex]
We plug in [tex]\sqrt{2x+1[/tex] into [tex]g(x)=x^2-4[/tex]
[tex](g\circ h)=(\sqrt{2x+1})^2-4[/tex]
[tex](g\circ h)=2x+1-4[/tex]
[tex](g\circ h)=2x-3[/tex]
QUESTION 3
i) Given [tex]g(x)=x+5[/tex] and [tex]h(x)=\sqrt{x-4}[/tex].
[tex](g\circ h)=g(h(x))[/tex]
[tex](g\circ h)=g(\sqrt{x-4})[/tex]
We plug in [tex]\sqrt{x-4}[/tex] into [tex]g(x)=x+5[/tex]
[tex](g\circ h)=\sqrt{(x-4)+5}[/tex]
ii) Given [tex]g(x)=x+5[/tex] and [tex]h(x)=\sqrt{x-4}[/tex].
[tex](g\circ g)=g(g(x))[/tex]
[tex](g\circ g)=g(x+5)[/tex]
We plug in [tex]x+5[/tex] into [tex]g(x)=x+5[/tex]
[tex](g\circ g)=(x+5)+5}[/tex]
[tex](g\circ g)=x+10}[/tex]
QUESTION 4
The given functions are;
[tex]f(x)=-2x+1[/tex] and [tex]g(x)=-x^2[/tex]
[tex]f\circ f=f(f(x))[/tex]
[tex]f\circ f=f(-2x+1)[/tex]
[tex]f\circ f=-2(-2x+1)+1)[/tex]
[tex]f\circ f=4x-2+1)[/tex]
[tex]f\circ f=4x-1)[/tex]
ii)
[tex]g\circ g=g(g(x))[/tex]
[tex]g\circ g=g(-x^2)[/tex]
[tex]g\circ g=-(x^2)^2[/tex]
[tex]g\circ g=-x^4[/tex]
QUESTION 5
[tex]f(x)=5x-2;g(x)=x^2;h(x)=-3x[/tex]
[tex]h\circ g\circ f=h\circ (g(f(x))[/tex]
[tex]h\circ g\circ f=h\circ (g(5x-2))[/tex]
[tex]h\circ g\circ f=h\circ ((5x-2)^2)[/tex]
[tex]h\circ g\circ f=h\circ (25x^2-20x+4)[/tex]
[tex]h\circ g\circ f=h(25x^2-20x+4)[/tex]
[tex]h\circ g\circ f=-3(25x^2-20x+4)[/tex]
[tex]h\circ g\circ f=-75x^2+60x-12)[/tex]
ii.
[tex]f\circ f\circ g=f\circ (f(g(x))[/tex]
[tex]f\circ f\circ g=f\circ (f(x^2)[/tex]
[tex]f\circ f\circ g=f\circ (5x^2-2)[/tex]
[tex]f\circ f\circ g=f(5x^2-2)[/tex]
[tex]f\circ f\circ g=5(5x^2-2)-2[/tex]
[tex]f\circ f\circ g=25x^2-10-2[/tex]
[tex]f\circ f\circ g=25x^2-12[/tex]
