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STEP ONE:

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)=
Response area
(f−g)(x)=
Response area
(f⋅g)(x)=
Response area
(fg)(x)=
Response area
Question 2
STEP TWO:

Identify/explain the domain (including any restrictions) of each of the combined functions (you found these in STEP ONE):

(f+g)(x)

(f−g)(x)

(f⋅g)(x)

(fg)(x)

(2 points)
0 / 10000 Word Limit
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1 answer

Let's begin by calculating each of the combined functions from the given functions \( f(x) = x^2 + 3x - 10 \) and \( g(x) = x - 2 \).

Step One: Finding Combined Functions

  1. Sum of functions: \( (f + g)(x) \) \[ (f + g)(x) = f(x) + g(x) = (x^2 + 3x - 10) + (x - 2) = x^2 + 4x - 12 \]

  2. Difference of functions: \( (f - g)(x) \) \[ (f - g)(x) = f(x) - g(x) = (x^2 + 3x - 10) - (x - 2) = x^2 + 2x - 8 \]

  3. Product of functions: \( (f \cdot g)(x) \) \[ (f \cdot g)(x) = f(x) \cdot g(x) = (x^2 + 3x - 10)(x - 2) \] To expand this, multiply each term in \( f(x) \) by \( g(x) \): \[ = x^2(x - 2) + 3x(x - 2) - 10(x - 2) \] \[ = x^3 - 2x^2 + 3x^2 - 6x - 10x + 20 = x^3 + x^2 - 16x + 20 \]

  4. Function composition: \( (fg)(x) \) Here, \( (fg)(x) \) generally implies the same as the product. Thus: \[ (fg)(x) = (f \cdot g)(x) = x^3 + x^2 - 16x + 20 \]

Summary of Combined Functions:

  • \( (f + g)(x) = x^2 + 4x - 12 \)
  • \( (f - g)(x) = x^2 + 2x - 8 \)
  • \( (f \cdot g)(x) = x^3 + x^2 - 16x + 20 \)
  • \( (fg)(x) = x^3 + x^2 - 16x + 20 \)

Step Two: Domain of Each Combined Function

  1. Domain of \( (f + g)(x) \)

    • The domain of a polynomial function is all real numbers. Thus, the domain is: \[ \text{Domain of } (f + g)(x): \mathbb{R} \quad (-\infty, \infty) \]

  2. Domain of \( (f - g)(x) \)

    • Similar to addition, the domain of this polynomial function is also all real numbers. Thus, the domain is: \[ \text{Domain of } (f - g)(x): \mathbb{R} \quad (-\infty, \infty) \]

  3. Domain of \( (f \cdot g)(x) \)

    • The product of polynomial functions is also a polynomial, which has a domain of all real numbers: \[ \text{Domain of } (f \cdot g)(x): \mathbb{R} \quad (-\infty, \infty) \]

  4. Domain of \( (fg)(x) \)

    • As noted, it represents the same as the product \( (f \cdot g)(x) \) and thus: \[ \text{Domain of } (fg)(x): \mathbb{R} \quad (-\infty, \infty) \]

In summary:

  • The domain of each function is all real numbers: \( \mathbb{R} \) or \( (-\infty, \infty) \) with no restrictions.