f°g(x) = f(g(x))
To fid it we need to replace x = g(x) into f(x) as follows:
[tex]\begin{gathered} f(x)=-x+1 \\ f(g(x))=-(2x^3-x)+1 \\ f\circ g(x)=-2x^3+x+1 \end{gathered}[/tex]
Since the obtained function is a polynomial, its domain and range are all real numbers.
g°f(x) = g(f(x))
To fid it we need to replace x = f(x) into g(x) as follows:
[tex]\begin{gathered} g(x)=2x^3-x \\ g(f(x))=2(-x+1)^3-(-x+1) \\ g\circ f(x)=2\lbrack(-x)^3+3\cdot(-x)^2\cdot1+3\cdot(-x)\cdot1^2+1^3\text{\rbrack+x}-1 \\ g\circ f(x)=2\lbrack-x^3+3x^2-3x+1^{}\text{\rbrack+x}-1 \\ g\circ f(x)=-2x^3+6x^2-6x+2\text{+x}-1 \\ g\circ f(x)=-2x^3+6x^2-5x+1 \end{gathered}[/tex]
Since the obtained function is a polynomial, its domain and range are all real numbers.
