Alright, let's tackle this step-by-step.
### Part A: Find [tex]\( f(3) \)[/tex] and [tex]\( g(3) \)[/tex]
We are given the functions [tex]\( f(x) = 5x + 35 \)[/tex] and [tex]\( g(x) = x + 7 \)[/tex].
1. To find [tex]\( f(3) \)[/tex]:
[tex]\[
f(3) = 5(3) + 35
\][/tex]
Multiply 5 by 3:
[tex]\[
5 \cdot 3 = 15
\][/tex]
Add 35 to 15:
[tex]\[
15 + 35 = 50
\][/tex]
So, [tex]\( f(3) = 50 \)[/tex].
2. To find [tex]\( g(3) \)[/tex]:
[tex]\[
g(3) = 3 + 7
\][/tex]
Add 3 and 7:
[tex]\[
3 + 7 = 10
\][/tex]
So, [tex]\( g(3) = 10 \)[/tex].
Thus, the values are:
[tex]\[
f(3) = 50 \quad \text{and} \quad g(3) = 10
\][/tex]
### Part B: Find [tex]\(\left(\frac{f}{g}\right)(3)\)[/tex]
We need to find the value of the function [tex]\(\frac{f(x)}{g(x)}\)[/tex] at [tex]\( x = 3 \)[/tex].
Given:
[tex]\[
f(3) = 50 \quad \text{and} \quad g(3) = 10
\][/tex]
The function [tex]\(\frac{f(x)}{g(x)}\)[/tex] is:
[tex]\[
\frac{f(3)}{g(3)} = \frac{50}{10}
\][/tex]
Perform the division:
[tex]\[
\frac{50}{10} = 5.0
\][/tex]
So, [tex]\(\left(\frac{f}{g}\right)(3) = 5.0\)[/tex].
### Part C: What is the domain of [tex]\(\left(\frac{f}{g}\right)(x)\)[/tex]?
The domain of [tex]\(\frac{f(x)}{g(x)}\)[/tex] is determined by where [tex]\( g(x) \)[/tex] is not equal to zero since division by zero is undefined.
We have:
[tex]\[
g(x) = x + 7
\][/tex]
We need to find the values of [tex]\( x \)[/tex] for which [tex]\( g(x) \neq 0 \)[/tex]:
[tex]\[
x + 7 \neq 0
\][/tex]
Subtract 7 from both sides:
[tex]\[
x \neq -7
\][/tex]
Therefore, the domain of [tex]\(\frac{f(x)}{g(x)}\)[/tex] is:
[tex]\[
\text{All real numbers except } x = -7
\][/tex]
In summary:
- [tex]\( f(3) = 50 \)[/tex]
- [tex]\( g(3) = 10 \)[/tex]
- [tex]\(\left(\frac{f}{g}\right)(3) = 5.0 \)[/tex]
- The domain of [tex]\(\left(\frac{f}{g}\right)(x)\)[/tex] is all real numbers except [tex]\( x = -7 \)[/tex].
