Sure, let's work through the simplification of the function [tex]\( f(x) \)[/tex] step by step. The function is given by:
[tex]\[
f(x) = \frac{x^2 + 2x - 48}{x^2 - 10x + 9}
\][/tex]
### Step 1: Factor the numerator and the denominator
First, we will factor both the numerator [tex]\(x^2 + 2x - 48\)[/tex] and the denominator [tex]\(x^2 - 10x + 9\)[/tex].
#### Factor the numerator [tex]\(x^2 + 2x - 48\)[/tex]:
We are looking for two numbers that multiply to [tex]\(-48\)[/tex] and add up to [tex]\(2\)[/tex].
The numbers that work are [tex]\(8\)[/tex] and [tex]\(-6\)[/tex]. So, we can write:
[tex]\[
x^2 + 2x - 48 = (x + 8)(x - 6)
\][/tex]
#### Factor the denominator [tex]\(x^2 - 10x + 9\)[/tex]:
We are looking for two numbers that multiply to [tex]\(9\)[/tex] and add up to [tex]\(-10\)[/tex].
The numbers that work are [tex]\(-1\)[/tex] and [tex]\(-9\)[/tex]. So, we can write:
[tex]\[
x^2 - 10x + 9 = (x - 1)(x - 9)
\][/tex]
### Step 2: Rewrite the function using the factorized forms
Now we will rewrite the function [tex]\(f(x)\)[/tex] using the factored forms of the numerator and the denominator:
[tex]\[
f(x) = \frac{(x + 8)(x - 6)}{(x - 1)(x - 9)}
\][/tex]
### Step 3: Simplify the function (if possible)
Check for any common factors in the numerator and the denominator that can be cancelled out. In this case, there are no common factors between [tex]\((x + 8)(x - 6)\)[/tex] and [tex]\((x - 1)(x - 9)\)[/tex].
Thus, the simplified form of the function remains:
[tex]\[
f(x) = \frac{(x + 8)(x - 6)}{(x - 1)(x - 9)}
\][/tex]
So, the function [tex]\(f(x)\)[/tex] is:
[tex]\[
f(x)=\frac{x^2 + 2x - 48}{x^2 - 10x + 9}
\][/tex]
