Question

Part AGiven f (x) = -x + 1 and g(x) = 2x - x, find [fºg)(x) and (sf(x). State the domain and range for each.Find [f.g] (x) and (gof)(x).Drag the expressions to the correct bins to define each function.[fogl(x)[g • f](x)-21' to+ 121 - 21+1-21 -+1-21 -+221²-51+ 2- 21+ + 21+--21 +6r²-51+1

Answer 1

f°g(x) = f(g(x))

To fid it we need to replace x = g(x) into f(x) as follows:

`\begin{gathered} f(x)=-x+1 \\ f(g(x))=-(2x^3-x)+1 \\ f\circ g(x)=-2x^3+x+1 \end{gathered}`

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:

`\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}`

Since the obtained function is a polynomial, its domain and range are all real numbers.