Answer :
Given
[tex]\begin{gathered} f(x)=-2x-1 \\ g(x)=-\frac{1}{2}x+\frac{1}{2} \end{gathered}[/tex]To find:
Which statements, if any, are true about these functions?
I. The function f(g(x)) = x for all real x.
Il. The function g(f(x)) = x for all real x.
Ill. Functions fand g are inverse functions.
Explanation:
It is given that,
[tex]\begin{gathered} f(x)=-2x-1 \\ g(x)=-\frac{1}{2}x+\frac{1}{2} \end{gathered}[/tex]Then,
[tex]\begin{gathered} f(g(x))=f(-\frac{1}{2}x+\frac{1}{2}) \\ =-2(-\frac{1}{2}x+\frac{1}{2})-1 \\ =x-1-1 \\ =x-2 \end{gathered}[/tex]Hence, statement I is not true.
Also,
[tex]\begin{gathered} g(f(x))=g(-2x-1)+\frac{1}{2} \\ =-\frac{1}{2}(-2x-1)+\frac{1}{2} \\ =x+\frac{1}{2}+\frac{1}{2} \\ =x+1 \end{gathered}[/tex]Hence, statement II is not true.
And, since
[tex]f(g(x))\ne g(f(x))[/tex]Then, f and g are not inverse to each other.
Hence, the answer is d) none of the statements are true.