To determine when the function [tex]\( f(x) = \frac{1}{x} \)[/tex] is undefined, we need to consider the denominator of the fraction.
1. The function [tex]\( f(x) \)[/tex] is a rational function, and a rational function is undefined whenever its denominator is zero.
2. For [tex]\( f(x) = \frac{1}{x} \)[/tex], the denominator is [tex]\( x \)[/tex].
3. Setting the denominator equal to zero to find the points where the function is undefined:
[tex]\[
x = 0
\][/tex]
4. Thus, [tex]\( f(x) = \frac{1}{x} \)[/tex] is undefined when [tex]\( x = 0 \)[/tex].
Now, let’s consider the function [tex]\( g(x) = x + 4 \)[/tex]. This is a linear function of the form [tex]\( g(x) = mx + b \)[/tex] where [tex]\( m \)[/tex] and [tex]\( b \)[/tex] are constants (in this case, [tex]\( m = 1 \)[/tex] and [tex]\( b = 4 \)[/tex]).
1. Linear functions are defined for all real numbers.
2. Therefore, the function [tex]\( g(x) = x + 4 \)[/tex] does not have any restrictions on its domain.
Conclusion:
The function [tex]\( f(x) = \frac{1}{x} \)[/tex] is undefined when [tex]\( x = 0 \)[/tex], while the function [tex]\( g(x) = x + 4 \)[/tex] is defined for all real numbers. Therefore, [tex]\( f(x) \)[/tex] has a restricted domain at [tex]\( x = 0 \)[/tex].
