Question

For each pair of funcation f and g below find f

Answer 1

(a)

Given

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

Then,

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

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.