To solve the given problem of determining the function [tex]\( F(x) \)[/tex], we need to express it in its simplest form. The function provided is:
[tex]\[ F(x) = \frac{3x + 5}{7x - 2} \][/tex]
Here is the explanation on how to understand and simplify the function step-by-step:
1. Identify the components of the function.
- The numerator of the function is [tex]\( 3x + 5 \)[/tex].
- The denominator of the function is [tex]\( 7x - 2 \)[/tex].
2. Simplify the fraction if possible.
- We're looking for any common factors in the numerator and the denominator that we can cancel out. However, upon inspection, there aren't any common factors between [tex]\( 3x + 5 \)[/tex] and [tex]\( 7x - 2 \)[/tex].
3. Domain of the function.
- The domain of the function is all real numbers except for values of [tex]\( x \)[/tex] that make the denominator zero, as division by zero is undefined.
- To find these values, set the denominator equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[ 7x - 2 = 0 \][/tex]
[tex]\[ 7x = 2 \][/tex]
[tex]\[ x = \frac{2}{7} \][/tex]
- Therefore, the function [tex]\( F(x) \)[/tex] is undefined at [tex]\( x = \frac{2}{7} \)[/tex].
4. Final representation.
- Since there are no further simplifications needed, the function remains:
[tex]\[ F(x) = \frac{3x + 5}{7x - 2} \][/tex]
In conclusion, the function [tex]\( F(x) = \frac{3x + 5}{7x - 2} \)[/tex] is already in its simplest and most interpretable form, with the domain being all real numbers except [tex]\( x = \frac{2}{7} \)[/tex]. This provides the complete representation and understanding of the given function.
