To find the function [tex]\(\left(\frac{f}{g}\right)(x)\)[/tex] given [tex]\( f(x) = x^2 - 9 \)[/tex] and [tex]\( g(x) = x^2 - 7x + 12 \)[/tex], let's proceed with division of the two functions.
First, let's write down the functions explicitly:
[tex]\[ f(x) = x^2 - 9 \][/tex]
[tex]\[ g(x) = x^2 - 7x + 12 \][/tex]
We need to find:
[tex]\[ h(x) = \frac{f(x)}{g(x)} = \frac{x^2 - 9}{x^2 - 7x + 12} \][/tex]
To simplify this expression, we can try factoring both the numerator and the denominator.
Step 1: Factorize the numerator [tex]\( f(x) \)[/tex]:
[tex]\[ f(x) = x^2 - 9 \][/tex]
Notice that [tex]\( x^2 - 9 \)[/tex] is a difference of squares, which can be factored as:
[tex]\[ f(x) = (x - 3)(x + 3) \][/tex]
Step 2: Factorize the denominator [tex]\( g(x) \)[/tex]:
[tex]\[ g(x) = x^2 - 7x + 12 \][/tex]
We need to find two numbers that multiply to 12 and add up to -7. These numbers are -3 and -4. Therefore, the factorization is:
[tex]\[ g(x) = (x - 3)(x - 4) \][/tex]
Step 3: Simplify the rational expression:
[tex]\[ \frac{f(x)}{g(x)} = \frac{(x - 3)(x + 3)}{(x - 3)(x - 4)} \][/tex]
We see that the factor [tex]\( x - 3 \)[/tex] appears in both the numerator and the denominator. Assuming [tex]\( x \neq 3 \)[/tex] (to avoid division by zero), we can cancel these terms:
[tex]\[ \frac{(x - 3)(x + 3)}{(x - 3)(x - 4)} = \frac{x + 3}{x - 4} \][/tex]
Hence, the simplified function is:
[tex]\[ h(x) = \frac{x + 3}{x - 4} \][/tex]
Thus, the result of the function [tex]\(\left(\frac{f}{g}\right)(x)\)[/tex] is:
[tex]\[ \left(\frac{f}{g}\right)(x) = \frac{x + 3}{x - 4} \][/tex]
