Questions: Find the x-intercept(s) of the following rational function: f(x)=(x+4)/(x+9)

Solution Steps

To find the $x$-intercept(s) of the rational function \( f(x) = \frac{x+4}{x+9} \), we need to determine the values of \( x \) for which \( f(x) = 0 \). This occurs when the numerator of the rational function is zero, as long as the denominator is not zero at the same point.

1. Set the numerator equal to zero and solve for \( x \).
2. Ensure that the denominator is not zero at the same \( x \) value.

To find the \( x \)-intercept(s) of the function \( f(x) = \frac{x+4}{x+9} \), we set the numerator equal to zero: \[ x + 4 = 0 \]

Solving the equation from Step 1 gives: \[ x = -4 \]

Next, we need to ensure that the denominator does not equal zero at \( x = -4 \): \[ x + 9 \neq 0 \implies -4 + 9 = 5 \neq 0 \] Since the denominator is not zero, \( x = -4 \) is a valid \( x \)-intercept.

Final Answer