Question

For each pair of funcation f and g below find f

Answer 1

(a)

Given

`\begin{gathered} f(x)=2x+3 \\ \text{and} \\ g(x)=(x-3)/(2) \end{gathered}`

Then,

`\begin{gathered} \Rightarrow f(g(x))=f((x-3)/(2))=2((x-3)/(2))+3=x-3+3=x \\ \Rightarrow f(g(x))=x \\ \text{and} \\ \Rightarrow g(f(x))=g(2x+3)=((2x+3)-3)/(2)=(2x)/(2)=x \\ \Rightarrow g(f(x))=x \end{gathered}`

f(g(x))=x, and g(f(x))=x

Therefore, f and g are inverses of each other.

(b)

Given

`\begin{gathered} f(x)=2x \\ \text{and} \\ g(x)=2x \end{gathered}`

Then,

`\begin{gathered} f(g(x))=f(2x)=2(2x)=4x \\ \Rightarrow f(g(x))=4x \\ \text{and} \\ g(f(x))=g(2x)=2(2x)=4x \\ \Rightarrow g(f(x))=4x \end{gathered}`

Therefore, f(g(x))=4x, and g(f(x))=4x

f and g are not inverses of each other.