Certainly! Let's analyze and solve each part of this problem step-by-step.
We are given two functions:
[tex]\[ f(x) = x^2 - 5 \][/tex]
[tex]\[ g(x) = 5x + 4 \][/tex]
### a. [tex]\((f + g)(x)\)[/tex]
We are tasked with finding the sum of the two functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex].
[tex]\[ (f + g)(x) = f(x) + g(x) \][/tex]
Substituting the given functions:
[tex]\[ (f + g)(x) = (x^2 - 5) + (5x + 4) \][/tex]
Combining like terms:
[tex]\[ (f + g)(x) = x^2 + 5x - 1 \][/tex]
### c. [tex]\((f - g)(x)\)[/tex]
Next, we find the difference between the two functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex].
[tex]\[ (f - g)(x) = f(x) - g(x) \][/tex]
Substituting the given functions:
[tex]\[ (f - g)(x) = (x^2 - 5) - (5x + 4) \][/tex]
Distributing the negative sign and combining like terms:
[tex]\[ (f - g)(x) = x^2 - 5x - 9 \][/tex]
### b. [tex]\((f + g)(3)\)[/tex]
We need to evaluate the sum of the functions at [tex]\(x = 3\)[/tex].
First, recall the expression for [tex]\((f + g)(x)\)[/tex]:
[tex]\[ (f + g)(x) = x^2 + 5x - 1 \][/tex]
Now, substitute [tex]\(x = 3\)[/tex]:
[tex]\[ (f + g)(3) = 3^2 + 5 \cdot 3 - 1 \][/tex]
[tex]\[ (f + g)(3) = 9 + 15 - 1 \][/tex]
[tex]\[ (f + g)(3) = 23 \][/tex]
### d. [tex]\((f - g)(5)\)[/tex]
Finally, we need to evaluate the difference of the functions at [tex]\(x = 5\)[/tex].
First, recall the expression for [tex]\((f - g)(x)\)[/tex]:
[tex]\[ (f - g)(x) = x^2 - 5x - 9 \][/tex]
Now, substitute [tex]\(x = 5\)[/tex]:
[tex]\[ (f - g)(5) = 5^2 - 5 \cdot 5 - 9 \][/tex]
[tex]\[ (f - g)(5) = 25 - 25 - 9 \][/tex]
[tex]\[ (f - g)(5) = -9 \][/tex]
### Summary
Summarizing the results, we get:
1. [tex]\((f + g)(x) = x^2 + 5x - 1\)[/tex]
2. [tex]\((f - g)(x) = x^2 - 5x - 9\)[/tex]
3. [tex]\((f + g)(3) = 23\)[/tex]
4. [tex]\((f - g)(5) = -9\)[/tex]
These are the detailed solutions for each question part.
