Questions: Simplify the function. f(x)=(x^2+4 x-21)/(x^2-9) f(x)= List any restrictions on the domain. (Enter your answers as a comma-separated list.) x ≠

Solution Steps

We start with the function

\[ f(x) = \frac{x^{2} + 4x - 21}{x^{2} - 9} \]

We factor the numerator \(x^{2} + 4x - 21\) and the denominator \(x^{2} - 9\):

• The numerator factors to \((x - 3)(x + 7)\).
• The denominator factors to \((x - 3)(x + 3)\).

Next, we simplify the function by canceling the common factor \((x - 3)\):

\[ f(x) = \frac{(x - 3)(x + 7)}{(x - 3)(x + 3)} = \frac{x + 7}{x + 3} \quad \text{for } x \neq 3 \]

To find the restrictions on the domain, we set the denominator equal to zero:

\[ x^{2} - 9 = 0 \implies (x - 3)(x + 3) = 0 \]

This gives us the solutions \(x = 3\) and \(x = -3\). Therefore, the restrictions on the domain are:

\[ x \neq -3, \quad x \neq 3 \]

Final Answer