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)=(1)/(3)x`

Required:

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

Step-by-step explanation:

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

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