Question

f(x) = \frac{x^2 - 7x + 12}{x^2 + 7x + 12}

Explanation:

To analyze the function \( f(x) = \frac{x^2 - 7x + 12}{x^2 + 7x + 12} \), we can factor the numerator and the denominator.

Step 1: Factor the numerator

The numerator is \( x^2 - 7x + 12 \). We need two numbers that multiply to \( 12 \) and add up to \( -7 \). These numbers are \( -3 \) and \( -4 \). So,
\[
x^2 - 7x + 12 = (x - 3)(x - 4)
\]

Step 2: Factor the denominator

The denominator is \( x^2 + 7x + 12 \). We need two numbers that multiply to \( 12 \) and add up to \( 7 \). These numbers are \( 3 \) and \( 4 \). So,
\[
x^2 + 7x + 12 = (x + 3)(x + 4)
\]

Step 3: Simplify the function

After factoring, the function becomes:
\[
f(x) = \frac{(x - 3)(x - 4)}{(x + 3)(x + 4)}
\]

We can also find the domain of the function. The denominator cannot be zero, so:
\[
(x + 3)(x + 4)
eq 0 \implies x
eq -3 \text{ and } x
eq -4
\]

If we want to find the \( x \)-intercepts, we set the numerator equal to zero:
\[
(x - 3)(x - 4) = 0 \implies x = 3 \text{ or } x = 4
\]

If we want to find the \( y \)-intercept, we set \( x = 0 \):
\[
f(0) = \frac{0^2 - 7(0) + 12}{0^2 + 7(0) + 12} = \frac{12}{12} = 1
\]

Final simplified form

The simplified form of the function is:
\[
f(x) = \frac{(x - 3)(x - 4)}{(x + 3)(x + 4)}, \quad x
eq -3, -4
\]

Answer:

To analyze the function \( f(x) = \frac{x^2 - 7x + 12}{x^2 + 7x + 12} \), we can factor the numerator and the denominator.

Step 1: Factor the numerator

The numerator is \( x^2 - 7x + 12 \). We need two numbers that multiply to \( 12 \) and add up to \( -7 \). These numbers are \( -3 \) and \( -4 \). So,
\[
x^2 - 7x + 12 = (x - 3)(x - 4)
\]

Step 2: Factor the denominator

The denominator is \( x^2 + 7x + 12 \). We need two numbers that multiply to \( 12 \) and add up to \( 7 \). These numbers are \( 3 \) and \( 4 \). So,
\[
x^2 + 7x + 12 = (x + 3)(x + 4)
\]

Step 3: Simplify the function

After factoring, the function becomes:
\[
f(x) = \frac{(x - 3)(x - 4)}{(x + 3)(x + 4)}
\]

We can also find the domain of the function. The denominator cannot be zero, so:
\[
(x + 3)(x + 4)
eq 0 \implies x
eq -3 \text{ and } x
eq -4
\]

If we want to find the \( x \)-intercepts, we set the numerator equal to zero:
\[
(x - 3)(x - 4) = 0 \implies x = 3 \text{ or } x = 4
\]

If we want to find the \( y \)-intercept, we set \( x = 0 \):
\[
f(0) = \frac{0^2 - 7(0) + 12}{0^2 + 7(0) + 12} = \frac{12}{12} = 1
\]

Final simplified form

The simplified form of the function is:
\[
f(x) = \frac{(x - 3)(x - 4)}{(x + 3)(x + 4)}, \quad x
eq -3, -4
\]