Answer:
The first number is [tex]0[/tex].
The second number is [tex]9[/tex].
Step-by-step explanation:
The circle "[tex]\circ[/tex]" in [tex]f \circ g[/tex] is the function composition operator. As its name suggests, it combines two functions, [tex]f[/tex] and [tex]g[/tex], to give a new function.
In practice, [tex](f\circ g)(x)[/tex] is equivalent to [tex]f(g(x))[/tex]. To calculate
In this case, [tex]\displaystyle f(x) = \frac{1}{x}[/tex] and [tex]g(x) = x^2 - 9x[/tex].
To find an expression for [tex](f\circ g)(x)[/tex] or [tex]f(g(x))[/tex], simply replace all occurrences of [tex]x[/tex] in [tex]f[/tex] with [tex]g(x)[/tex]. That is:
[tex](f \circ g)(x) = f(g(x)) = \displaystyle \frac{1}{g(x)}[/tex].
Since [tex]g(x) = x^2 - 9x[/tex], [tex]\displaystyle \frac{1}{g(x)} = \frac{1}{x^2 - 9x}[/tex].
Hence, [tex]\displaystyle (f \circ g) (x) = \frac{1}{x^2 - 9x}[/tex].
The expression [tex]\displaystyle (f \circ g) (x) = \frac{1}{x^2 - 9x}[/tex] is undefined only when the denominator [tex]\left(x^2 - 9x\right)[/tex] is equal to zero. That is: [tex]x^2 - 9x = 0[/tex].
Factor [tex]x[/tex] out to obtain: [tex]x\cdot ( x - 9) = 0[/tex].
That happens either when [tex]x = 0[/tex] or when [tex]x = 9[/tex].
