Question

Given a function $f\left(x\right)=\frac{1}{2}x-4$ and the composite function $f\left(g\right(x\left)\right)=g\left(f\right(x\left)\right)$, determine which among the following can be $g\left(x\right)$:
I. $2x-\frac{1}{4}$
II. $2x+8$
III. $\frac{1}{2}x-4$

• A. I only
• B. II only
• C. III only
• D. II and III only
• E. I, II, and III

Answer 1

The correct option is D II and III only

If $g\left(x\right)=2x-\frac{1}{4}$

$g\left(f\right(x\left)\right)=x-8-\frac{1}{4}$

$f\left(g\right(x\left)\right)=x-\frac{1}{8}-4$

Case II

$g\left(f\right(x\left)\right)=2(\frac{x}{2}-4)+8=x$

$f\left(g\right(x\left)\right)=\frac{1}{2}(2x+8)-4=x+4-4=x$

Case III

$g\left(f\right(x\left)\right)=\frac{1}{2}\left(\frac{x-8}{2}\right)-4=\frac{x}{4}-6$

$f\left(g\right(x\left)\right)=\frac{1}{2}\left(\frac{x-8}{2}\right)-4=\frac{x}{4}-6$

Option D is correct