To find \((f+g)(x)\), we need to add the two functions \(f(x)\) and \(g(x)\) together. Here's the step-by-step process to obtain \((f+g)(x)\):
Given:
[tex]\[
f(x) = 4x^2 + 5x - 3
\][/tex]
[tex]\[
g(x) = 4x^3 - 3x^2 + 5
\][/tex]
To find \((f+g)(x)\), we need to add \(f(x)\) and \(g(x)\):
[tex]\[
(f+g)(x) = f(x) + g(x)
\][/tex]
Let's substitute \(f(x)\) and \(g(x)\) with their respective expressions:
[tex]\[
(f+g)(x) = (4x^2 + 5x - 3) + (4x^3 - 3x^2 + 5)
\][/tex]
Now, combine like terms:
1. The \(x^3\) term:
[tex]\[
4x^3
\][/tex]
2. The \(x^2\) terms:
[tex]\[
4x^2 - 3x^2 = x^2
\][/tex]
3. The \(x\) term:
[tex]\[
5x
\][/tex]
4. The constant terms:
[tex]\[
-3 + 5 = 2
\][/tex]
Putting it all together, we get:
[tex]\[
(f+g)(x) = 4x^3 + x^2 + 5x + 2
\][/tex]
So, \((f+g)(x)\) is:
[tex]\[
4x^3 + x^2 + 5x + 2
\][/tex]
If you want to find the value of \((f+g)(x)\) at \(x = 1\):
Substitute \(x = 1\) into \((f+g)(x)\):
[tex]\[
(f+g)(1) = 4(1)^3 + (1)^2 + 5(1) + 2
\][/tex]
Calculate the value step-by-step:
[tex]\[
= 4(1) + 1 + 5 + 2
\][/tex]
[tex]\[
= 4 + 1 + 5 + 2
\][/tex]
[tex]\[
= 12
\][/tex]
Therefore, \((f+g)(1)\) is:
[tex]\[
12
\][/tex]
