Asked by BlUbArRy!2#
Given f(x)=x2+3x−10
and g(x)=x−2
find each of the following combined functions.
NOTE: You will NOT use each response option.
(1 point)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
(f+g)(x)=
(f−g)(x)=
(f⋅g)(x)=
Response area
and g(x)=x−2
find each of the following combined functions.
NOTE: You will NOT use each response option.
(1 point)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
(f+g)(x)=
(f−g)(x)=
(f⋅g)(x)=
Response area
Answers
Answered by
GPT-4o mini
To find the combined functions (f+g)(x), (f−g)(x), and (f⋅g)(x) given the functions:
- \( f(x) = x^2 + 3x - 10 \)
- \( g(x) = x - 2 \)
we perform the following calculations:
1. **(f + g)(x)**:
\[
(f + g)(x) = f(x) + g(x) = (x^2 + 3x - 10) + (x - 2)
\]
Combine like terms:
\[
= x^2 + 3x + x - 10 - 2 = x^2 + 4x - 12
\]
2. **(f - g)(x)**:
\[
(f - g)(x) = f(x) - g(x) = (x^2 + 3x - 10) - (x - 2)
\]
Combine like terms:
\[
= x^2 + 3x - 10 - x + 2 = x^2 + 2x - 8
\]
3. **(f ⋅ g)(x)**:
\[
(f ⋅ g)(x) = f(x) \cdot g(x) = (x^2 + 3x - 10)(x - 2)
\]
Expand using the distributive property:
\[
= x^2(x - 2) + 3x(x - 2) - 10(x - 2)
\]
\[
= x^3 - 2x^2 + 3x^2 - 6x - 10x + 20
\]
Combine like terms:
\[
= x^3 + (3x^2 - 2x^2) + (-6x - 10x) + 20 = x^3 + x^2 - 16x + 20
\]
Thus, the combined functions are:
- \((f + g)(x) = x^2 + 4x - 12\)
- \((f - g)(x) = x^2 + 2x - 8\)
- \((f \cdot g)(x) = x^3 + x^2 - 16x + 20\)
You can input these results into the appropriate fields for your assignment.
- \( f(x) = x^2 + 3x - 10 \)
- \( g(x) = x - 2 \)
we perform the following calculations:
1. **(f + g)(x)**:
\[
(f + g)(x) = f(x) + g(x) = (x^2 + 3x - 10) + (x - 2)
\]
Combine like terms:
\[
= x^2 + 3x + x - 10 - 2 = x^2 + 4x - 12
\]
2. **(f - g)(x)**:
\[
(f - g)(x) = f(x) - g(x) = (x^2 + 3x - 10) - (x - 2)
\]
Combine like terms:
\[
= x^2 + 3x - 10 - x + 2 = x^2 + 2x - 8
\]
3. **(f ⋅ g)(x)**:
\[
(f ⋅ g)(x) = f(x) \cdot g(x) = (x^2 + 3x - 10)(x - 2)
\]
Expand using the distributive property:
\[
= x^2(x - 2) + 3x(x - 2) - 10(x - 2)
\]
\[
= x^3 - 2x^2 + 3x^2 - 6x - 10x + 20
\]
Combine like terms:
\[
= x^3 + (3x^2 - 2x^2) + (-6x - 10x) + 20 = x^3 + x^2 - 16x + 20
\]
Thus, the combined functions are:
- \((f + g)(x) = x^2 + 4x - 12\)
- \((f - g)(x) = x^2 + 2x - 8\)
- \((f \cdot g)(x) = x^3 + x^2 - 16x + 20\)
You can input these results into the appropriate fields for your assignment.
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