To find the root of the function [tex]\( f(x) = \frac{1}{x+3} \)[/tex], we need to solve for [tex]\( x \)[/tex] such that [tex]\( f(x) = 0 \)[/tex].
First, let's set the function equal to zero and analyze the equation:
[tex]\[
f(x) = \frac{1}{x+3} = 0
\][/tex]
This means we need to solve:
[tex]\[
\frac{1}{x+3} = 0
\][/tex]
For a fraction to equal zero, its numerator must be zero. However, the numerator in our fraction is a constant 1, and it will never be equal to zero no matter what value [tex]\( x \)[/tex] takes. This fraction [tex]\( \frac{1}{x+3} \)[/tex] is undefined when the denominator is zero:
[tex]\[
x + 3 = 0 \\
x = -3
\][/tex]
At [tex]\( x = -3 \)[/tex], the function is undefined, not zero. Therefore, the function [tex]\( f(x) = \frac{1}{x+3} \)[/tex] does not have any real roots because there is no value of [tex]\( x \)[/tex] that can satisfy the equation [tex]\( \frac{1}{x+3} = 0 \)[/tex].
Given the choices:
A. none
B. [tex]\( x = -3 \)[/tex]
C. [tex]\( x = -2 \)[/tex]
D. [tex]\( x = -4 \)[/tex]
The correct answer is:
A. none
