Given the functions:

[tex]\[
\begin{array}{l}
f(x) = 2x^2 + 4x - 5 \\
g(x) = 6x^3 - 2x^2 + 3
\end{array}
\][/tex]

Find [tex]\((f - g)(x)\)[/tex].



Answer :

Certainly! Let's find \((f - g)(x)\) step-by-step.

Given the two functions:
[tex]\[ f(x) = 2x^2 + 4x - 5 \][/tex]
[tex]\[ g(x) = 6x^3 - 2x^2 + 3 \][/tex]

We need to find \((f - g)(x)\), which means we subtract \(g(x)\) from \(f(x)\).

Let's start by writing out the expression for \((f - g)(x)\):
[tex]\[ (f - g)(x) = f(x) - g(x) \][/tex]

Substitute the given functions into this expression:
[tex]\[ (f - g)(x) = (2x^2 + 4x - 5) - (6x^3 - 2x^2 + 3) \][/tex]

Now distribute the negative sign to each term in \(g(x)\):
[tex]\[ (f - g)(x) = 2x^2 + 4x - 5 - 6x^3 + 2x^2 - 3 \][/tex]

Next, combine like terms. Start with the highest degree term and proceed to the lowest:

1. The highest degree term is \( -6x^3 \), and it appears only in \(g(x)\):
[tex]\[ -6x^3 \][/tex]

2. For the \(x^2\) terms, we have \(2x^2\) from both \(f(x)\) and the subtracted \(g(x)\):
[tex]\[ 2x^2 + 2x^2 = 4x^2 \][/tex]

3. For the \(x\) terms, we have:
[tex]\[ 4x \][/tex]

4. Finally, the constant terms are \(-5\) from \(f(x)\) and \(-3\) from the subtracted \(g(x)\):
[tex]\[ -5 - 3 = -8 \][/tex]

Putting these all together, we get:
[tex]\[ (f - g)(x) = -6x^3 + 4x^2 + 4x - 8 \][/tex]

So, the result of \((f - g)(x)\) is:
[tex]\[ \boxed{-6x^3 + 4x^2 + 4x - 8} \][/tex]