To find the difference in the salary Cynthia will earn working as a camp counselor instead of a lifeguard after [tex]$x$[/tex] hours, we need to compute the difference between the functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]. The functions are given as:
[tex]\[ f(x) = 10x + 50 \][/tex]
[tex]\[ g(x) = 15x + 25 \][/tex]
The operation [tex]\((f - g)(x)\)[/tex] represents the difference between the earnings from the two jobs. To find this, we subtract [tex]\( g(x) \)[/tex] from [tex]\( f(x) \)[/tex]:
[tex]\[ (f - g)(x) = f(x) - g(x) \][/tex]
Substituting the given expressions for [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
[tex]\[ (f - g)(x) = (10x + 50) - (15x + 25) \][/tex]
Next, we distribute the negative sign through the terms inside the parentheses:
[tex]\[ (f - g)(x) = 10x + 50 - 15x - 25 \][/tex]
Then, we combine like terms. Combine the terms containing [tex]\( x \)[/tex]:
[tex]\[ 10x - 15x = -5x \][/tex]
And combine the constant terms:
[tex]\[ 50 - 25 = 25 \][/tex]
Putting it all together, we get:
[tex]\[ (f - g)(x) = -5x + 25 \][/tex]
Thus, the function [tex]\((f - g)(x)\)[/tex] representing the difference in salary Cynthia will earn after [tex]\( x \)[/tex] hours by working as a camp counselor instead of a lifeguard is:
[tex]\[ (f - g)(x) = -5x + 25 \][/tex]
So, the correct answer is:
[tex]\[ \boxed{(f - g)(x) = -5x + 25} \][/tex]
