Question

Consider these functions:-1l)=-号5十号Which statements, if any, are true about these functions?I. The function (e(x)) = × for all real x.Il. The function g(x) = × for all real x.Ill. Functions fand g are inverse functions.

Answer 1

Given

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

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.

Step-by-step explanation:

It is given that,

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

Then,

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

Hence, statement I is not true.

Also,

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

Hence, statement II is not true.

And, since

`f(g(x))\\e g(f(x))`

Then, f and g are not inverse to each other.

Hence, the answer is d) none of the statements are true.