Questions: h(x)=(x^2-13x+36)/(x^2-6x+8)

Solution Steps

To analyze the function \( h(x) = \frac{x^{2} - 13x + 36}{x^{2} - 6x + 8} \), we need to identify the points where the function is undefined and simplify the expression if possible. The function is undefined where the denominator is zero. We can find these points by solving the equation \( x^{2} - 6x + 8 = 0 \). Additionally, we can simplify the function by factoring both the numerator and the denominator and canceling any common factors.

The function \( h(x) = \frac{x^2 - 13x + 36}{x^2 - 6x + 8} \) is undefined where the denominator is zero. To find these points, solve the equation:

\[ x^2 - 6x + 8 = 0 \]

Factoring the quadratic equation, we get:

\[ (x - 2)(x - 4) = 0 \]

Thus, the function is undefined at \( x = 2 \) and \( x = 4 \).

Next, we simplify the function by factoring both the numerator and the denominator:

• The numerator \( x^2 - 13x + 36 \) factors to \( (x - 9)(x - 4) \).
• The denominator \( x^2 - 6x + 8 \) factors to \( (x - 2)(x - 4) \).

The function simplifies to:

\[ h(x) = \frac{(x - 9)(x - 4)}{(x - 2)(x - 4)} = \frac{x - 9}{x - 2} \]

Note that the factor \( (x - 4) \) cancels out, but \( x = 4 \) is still a point of discontinuity due to the original denominator.

Final Answer