To find the function that represents the voltage in the given circuit, we need to follow these steps:
1. Identify the parameters:
- Current ([tex]\(I\)[/tex]) is 2 Amperes.
- Resistance ([tex]\(R\)[/tex]) is represented by the function [tex]\(\frac{x^2 - x - 30}{2x - 12}\)[/tex] Ohms.
2. Simplify the resistance function [tex]\(R\)[/tex]:
[tex]\[
R = \frac{x^2 - x - 30}{2x - 12}
\][/tex]
First, factorize the numerator [tex]\(x^2 - x - 30\)[/tex]:
[tex]\[
x^2 - x - 30 = (x - 6)(x + 5)
\][/tex]
This allows us to rewrite the resistance function as:
[tex]\[
R = \frac{(x - 6)(x + 5)}{2x - 12}
\][/tex]
Notice that the denominator [tex]\(2x - 12\)[/tex] can be factored out as:
[tex]\[
2x - 12 = 2(x - 6)
\][/tex]
Therefore, the resistance function can be further simplified by cancelling out [tex]\(x - 6\)[/tex] from the numerator and denominator:
[tex]\[
R = \frac{(x - 6)(x + 5)}{2(x - 6)} = \frac{x + 5}{2}
\][/tex]
So, the simplified resistance function [tex]\(R\)[/tex] is:
[tex]\[
R = \frac{x}{2} + \frac{5}{2}
\][/tex]
3. Calculate the voltage [tex]\(V\)[/tex] using the formula [tex]\(V = I \times R\)[/tex]:
Since the current [tex]\(I\)[/tex] is 2 Amperes, substitute [tex]\(I = 2\)[/tex] into the voltage formula:
[tex]\[
V = 2 \times \left(\frac{x}{2} + \frac{5}{2}\right)
\][/tex]
Distribute the 2:
[tex]\[
V = 2 \times \frac{x}{2} + 2 \times \frac{5}{2}
\][/tex]
This simplifies to:
[tex]\[
V = x + 5
\][/tex]
So, the function that represents the voltage in the circuit is:
[tex]\[
\boxed{x + 5 \text{ Volts}}
\][/tex]
