Question

Let f (x) = 1/x
and g(x) = x² – 3x. What
two numbers are not in the domain of fºg?

Answer 1

0 and 3

Step-by-step explanation

The given functions are

`f(x) = (1)/(x)`

and

`g(x) = {x}^(2) - 3x`

`( f \circ g)(x) = f(g(x))`

`( f \circ g)(x) = f( {x}^(2) -x )`

`( f \circ g)(x) = \frac{1}{ {x}^(2) - 3x}`

Factor the numerator:

`( f \circ g)(x) = (1)/( x(x - 3))`

The function will be undefined if the denominator is zero.

`x(x - 3) \\e0`

`x \\e0 \: and \: x \\e3`

Therefore 0 and not in the domain of the composed function.

Answer 2

For this case we have the following equations:

`f (x) = \frac {1} {x}\\g (x) = x ^ 2-3x`

We must find
`(f_ {o} g) (x):`

By definition of composition of functions we have to:

`(f_ {o} g) (x) = f (g (x))`

So:

`(f_ {o} g) (x) = \frac {1} {x ^ 2-3x}`

We must find the domain of f (g (x)). The domain will be given by the values for which the function is defined, that is, when the denominator is nonzero.

`x ^ 2-3x = 0\\x (x-3) = 0`

So, the roots are:

`x_ {1} = 0\\x_ {2} = 3`

The domain is given by all real numbers except 0 and 3.

Answer:

x other than 0 and 3