Given f(x)=x2+3x−10

Question

and g(x)=x−2
find each of the following combined functions.

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(f+g)(x)=

(f−g)(x)=

(f⋅g)(x)=
Response area

Answer 1

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.