Sure, let's carefully break down the solution to this problem step-by-step.
We are given two functions:
[tex]\[ f(x) = 2x^2 - 3x - 2 \][/tex]
[tex]\[ g(x) = \frac{1}{x} \][/tex]
We need to determine [tex]\((f+g)(x)\)[/tex], which means we need to add these two functions together:
[tex]\[ (f+g)(x) = f(x) + g(x) \][/tex]
Substituting the given functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
[tex]\[ (f+g)(x) = (2x^2 - 3x - 2) + \frac{1}{x} \][/tex]
So, to find [tex]\((f+g)(x)\)[/tex] at a specific value, we substitute [tex]\(x = 1\)[/tex] into the combined function:
Substitute [tex]\(x = 1\)[/tex]:
[tex]\[ f(1) = 2(1)^2 - 3(1) - 2 = 2(1) - 3(1) - 2 = 2 - 3 - 2 = -3 \][/tex]
[tex]\[ g(1) = \frac{1}{1} = 1 \][/tex]
Now, add the results of [tex]\(f(1)\)[/tex] and [tex]\(g(1)\)[/tex]:
[tex]\[ (f+g)(1) = f(1) + g(1) = -3 + 1 = -2 \][/tex]
So, the value of [tex]\((f+g)(x)\)[/tex] when [tex]\( x = 1 \)[/tex] is:
[tex]\[ \boxed{-2.0} \][/tex]
