Find left-parenthesis f plus g right-parenthesis left-parenthesis x right-parenthesis when lower f left-parenthesis x right-parenthesis equals x cubed minus 2 x squared plus 1 and lower g left-parenthesis x right-parenthesis equals 4 x cubed minus 5 x plus 7.

Question

(1 point)
Responses

left-parenthesis lower f plus lower g right-parenthesis left-parenthesis x right-parenthesis equals 5 x cubed minus 7 x plus 8
Image with alt text: left-parenthesis lower f plus lower g right-parenthesis left-parenthesis x right-parenthesis equals 5 x cubed minus 7 x plus 8

left-parenthesis lower f plus lower g right-parenthesis left-parenthesis x right-parenthesis equals 5 x cubed minus 7 x squared plus 8
Image with alt text: left-parenthesis lower f plus lower g right-parenthesis left-parenthesis x right-parenthesis equals 5 x cubed minus 7 x squared plus 8

left-parenthesis lower f plus lower g right-parenthesis left-parenthesis x right-parenthesis equals 4 x superscript 6 baseline minus 2 x squared minus 5 x plus 8
Image with alt text: left-parenthesis lower f plus lower g right-parenthesis left-parenthesis x right-parenthesis equals 4 x superscript 6 baseline minus 2 x squared minus 5 x plus 8

left-parenthesis lower f plus lower g right-parenthesis left-parenthesis x right-parenthesis equals 5 x cubed minus 2 x squared minus 5 x plus 8

Answer 1

To find \((f + g)(x)\), we will add the functions \(f(x)\) and \(g(x)\) together.

Given:

• \(f(x) = x^3 - 2x^2 + 1\)
• \(g(x) = 4x^3 - 5x + 7\)

Now we add \(f(x)\) and \(g(x)\):

\[ (f + g)(x) = f(x) + g(x) \] \[ = (x^3 - 2x^2 + 1) + (4x^3 - 5x + 7) \]

Now, combine like terms:

1. For \(x^3\):

   • \(x^3 + 4x^3 = 5x^3\)

2. For \(x^2\):

   • \(-2x^2\) (there is no \(x^2\) term in \(g(x)\))

3. For \(x\):

   • \(-5x\) (there is no \(x\) term in \(f(x)\))

4. Constant terms:

   • \(1 + 7 = 8\)

Now, combining these results, we have:

\[ (f + g)(x) = 5x^3 - 2x^2 - 5x + 8 \]

Thus, the final result is:

\[ \boxed{(f + g)(x) = 5x^3 - 2x^2 - 5x + 8} \]

This corresponds to the option:

(f + g)(x) = 5 x cubed - 2 x squared - 5 x + 8.