Question

3. Find f(g(x)) & g(f(x)),if f(x) = 3x& g(x) = (1/3) xWhat do you notice about f(g(x)) &g(f(x))? What does that tell you about f(x)& g(x)?

Answer 1

Given:

$f(x)=3x,g(x)=\frac{1}{3}x$

Required:

We need to find f(g(x)) and g(f(x)).

Explanation:

$Substitute\text{ }g(x)=\frac{1}{3}x\text{ in the }f(g(x)).$$f(g(x))=f(\frac{1}{3}x)$

$Replace\text{ }x=\frac{1}{3}x\text{ in the equation }f(x)=3x.$$f(\frac{1}{3}x)=3(\frac{1}{3}x)$$f(\frac{1}{3}x)=x$$Substitute\text{ f}(\frac{1}{4}x)=x\text{ in the }f(g(x))=f(\frac{1}{3}x).$$f(g(x))=x$

$Substitute\text{ f}(x)=3x\text{ in the }g(f(x)).$$g(f(x))=g(3x)$$Replace\text{ }x=3x\text{ in the equation }g(x)=\frac{1}{3}x.$$g(3x)=\frac{1}{3}(3x)$$g(3x)=x$$Substitute\text{ }g(3x)=x\text{ in }the\text{ equation }g(f(x))=g(3x).$$\text{ }g(f(x))=x$$If\text{ }f(g(x))=g(f(x))=x\text{ then f and g are inverse functions.}$

Final answer:

$f(g(x))=x$$f(g(x))=x$$\text{f and g are inverse functions.}$