To determine the domain of the function [tex]\( f(x) = \frac{5x + 3}{x} \)[/tex], we need to identify all possible values of [tex]\( x \)[/tex] for which the function is defined.
1. Domain:
Since the function [tex]\( f(x) = \frac{5x + 3}{x} \)[/tex] involves division by [tex]\( x \)[/tex], [tex]\( x \)[/tex] must not be zero because division by zero is undefined. Therefore, the domain of the function is all real numbers except [tex]\( x = 0 \)[/tex].
Hence, the domain in interval notation is:
[tex]\[ (-\infty, 0) \cup (0, \infty) \][/tex]
2. [tex]\(x\)[/tex]-intercept(s):
To find the [tex]\( x \)[/tex]-intercept(s), we need to set the function [tex]\( f(x) \)[/tex] equal to zero and solve for [tex]\( x \)[/tex].
[tex]\[
f(x) = \frac{5x + 3}{x} = 0
\][/tex]
The function [tex]\( f(x) \)[/tex] is equal to zero when the numerator is zero, because for a fraction to be zero, its numerator must be zero (and the denominator must be non-zero).
[tex]\[
5x + 3 = 0
\][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
5x = -3
\][/tex]
[tex]\[
x = -\frac{3}{5}
\][/tex]
So, the [tex]\( x \)[/tex]-intercept is:
[tex]\[
\left( -\frac{3}{5}, 0 \right)
\][/tex]
Final Answer:
- The domain of the function is: [tex]\((- \infty, 0) \cup (0, \infty)\)[/tex]
- The [tex]\( x \)[/tex]-intercept is: [tex]\( \left( -\frac{3}{5}, 0 \right) \)[/tex]
Since I must select the correct choice:
A. The [tex]\( x \)[/tex]-intercept(s) is/are [tex]\( \left( -\frac{3}{5}, 0 \right) \)[/tex]
Thus, ensure you select option A and fill in the intercept as [tex]\( \left( -\frac{3}{5}, 0 \right) \)[/tex].
