Sure, let's break down the function [tex]\( f(x) \)[/tex] step-by-step.
We have the function:
[tex]\[ f(x) = \frac{3x + 12}{6x - 4} \][/tex]
### Step-by-Step Solution:
1. Identify the components of the function:
- The numerator is [tex]\( 3x + 12 \)[/tex].
- The denominator is [tex]\( 6x - 4 \)[/tex].
2. Simplify the numerator and the denominator separately (if possible):
- The numerator [tex]\( 3x + 12 \)[/tex] can be factored as:
[tex]\[ 3x + 12 = 3(x + 4) \][/tex]
- The denominator [tex]\( 6x - 4 \)[/tex] can be factored as:
[tex]\[ 6x - 4 = 2(3x - 2) \][/tex]
Combining these factorizations, we get:
[tex]\[
f(x) = \frac{3(x + 4)}{2(3x - 2)}
\][/tex]
3. Identify and discuss any restrictions on [tex]\( x \)[/tex]:
- The denominator [tex]\( 6x - 4 \)[/tex] must not be equal to zero because division by zero is undefined. Setting the denominator to zero to find the restriction:
[tex]\[
6x - 4 = 0
\][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
6x = 4 \implies x = \frac{4}{6} \implies x = \frac{2}{3}
\][/tex]
- Thus, [tex]\( x \neq \frac{2}{3} \)[/tex].
4. Write the simplified function with the restriction:
[tex]\[
f(x) = \frac{3(x + 4)}{2(3x - 2)}, \quad \text{for } x \neq \frac{2}{3}
\][/tex]
### Summary:
The function [tex]\( f(x) = \frac{3x + 12}{6x - 4} \)[/tex] simplifies to [tex]\( \frac{3(x + 4)}{2(3x - 2)} \)[/tex]. The domain of this function excludes [tex]\( x = \frac{2}{3} \)[/tex].
So we have:
[tex]\[ f(x) = \frac{3x + 12}{6x - 4}, \quad \text{with } \ x \neq \frac{2}{3} \][/tex]
