Determine whether fog( x ) = gof( x ) and Find the domain of fog( x ) and gof( x )

If `f\left( x ) = | x | and g( x ) = \frac{\sqrt x }{x}`

1. Determine whether `fog( x ) = gof( x )`
2. Find the domain of `fog( x ) and gof( x )`

Solution:

• Determine whether `fog( x ) = gof( x )`

                Here

`f( x ) = | x | and g( x ) = \frac{\sqrt x }{x}`

                First, we find 

`fog(x) = ?`

`fog(x) = f(g(x))`

`fog(x) = f(\frac{\sqrt x }{x})`

`fog(x) = | \frac{\sqrt x }{x} |`

`fog(x) = | \frac{\sqrt x }{\sqrt {x^2}}|`

`fog(x) = |\frac{\sqrt x}{\sqrt x \sqrt x }|`

`fog(x) = |\frac{1}{\sqrt x }|       -  -  -  -  -  - (1)`

                Now, we find 

`gof(x) = ?`

`gof(x) = g(f(x))`

`gof(x) = g(|x|)`

`gof(x) = \frac{\sqrt |x| }{|x|}`

`gof(x) = \frac{\sqrt |x| }{\sqrt |x| .\sqrt |x| }` 

`gof(x) = \frac{1}{\sqrt|x|}      -----------(2)`

                By (1) and (2) , we can prove that

`fog(x) \ne gof(x)`

• Find the domain of \(fog\left( x \right)\;and\;gof\left( x \right)

               Domain of `fog(x) = (0,\infty )`

               Domain of `gof(x) = ( - \infty ,0)U(0,\infty )`

FAQ

Q1: What does `\(f\circ g(x)\)` mean?

It means the composition of f and g, written as `\(f(g(x))\)`. You first apply g(x) and then apply f to the result.

Q2: Are composite functions commutative?

No, in general `\(f\circ g(x)\ne g\circ f(x)\)`. The order of application matters.

Q3: How do you find the domain of a composite function?

Start with the domain requirements of the inner function and then ensure the outputs of the inner function fit the input requirements of the outer function.

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