Determine whether fog( x ) = gof( x ) and Find the domain of fog( x ) and gof( x ) | Query Point Official

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.

Explore more about CALCULUS AND ANALYTICAL GEOMETRY in Mathematics Notes & MCQs.

Query Point Official – Smart Notes for Exams & Conceptual Learning