Question

Let f (x) =1/x

Answer 1

Answer:

The first number is $0$.

The second number is $9$.

Step-by-step explanation:

The circle "$\circ$" in $f \circ g$ is the function composition operator. As its name suggests, it combines two functions, $f$ and $g$, to give a new function.

In practice, $(f\circ g)(x)$ is equivalent to $f(g(x))$. To calculate

In this case, $\displaystyle f(x) = \frac{1}{x}$ and $g(x) = x^2 - 9x$.

To find an expression for $(f\circ g)(x)$ or $f(g(x))$, simply replace all occurrences of $x$ in $f$ with $g(x)$. That is:

$(f \circ g)(x) = f(g(x)) = \displaystyle \frac{1}{g(x)}$.

Since $g(x) = x^2 - 9x$, $\displaystyle \frac{1}{g(x)} = \frac{1}{x^2 - 9x}$.

Hence, $\displaystyle (f \circ g) (x) = \frac{1}{x^2 - 9x}$.

The expression $\displaystyle (f \circ g) (x) = \frac{1}{x^2 - 9x}$ is undefined only when the denominator $\left(x^2 - 9x\right)$ is equal to zero. That is: $x^2 - 9x = 0$.

Factor $x$ out to obtain: $x\cdot ( x - 9) = 0$.

That happens either when $x = 0$ or when $x = 9$.