QuestionFind the simplified expression of $g(x) - f(x)$ , where $f(x) = 5x + 3$ and $g(x) = x^2 - x + 1$ . Then, determine the domain of the simplified expression.

Studdy Solution

STEP 1

1. The function $f(x) = 5x + 3$ is a linear function.
2. The function $g(x) = x^2 - x + 1$ is a quadratic function.
3. The domain of a function is the set of all possible input values (x-values) for which the function is defined.
4. The domain of a polynomial function is all real numbers, unless otherwise restricted.

STEP 2

1. Perform the function operation $g(x) - f(x)$ .
2. Simplify the result of the function operation.
3. Determine the domain of the resulting function.

STEP 3

Perform the function operation $g(x) - f(x)$ .
$g(x) - f(x) = (x^2 - x + 1) - (5x + 3)$

STEP 4

Distribute the negative sign to the terms in $f(x)$ .
$g(x) - f(x) = x^2 - x + 1 - 5x - 3$

STEP 5

Combine like terms.
$g(x) - f(x) = x^2 - 6x - 2$

STEP 6

The simplified result of the function operation is:
$g(x) - f(x) = x^2 - 6x - 2$

STEP 7

Determine the domain of the resulting function.
Since the resulting function $g(x) - f(x) = x^2 - 6x - 2$ is a polynomial, its domain is all real numbers.
Therefore, the domain of $g(x) - f(x)$ is $\mathbb{R}$ , which means all real numbers.
The simplified expression for $g(x) - f(x)$ is $x^2 - 6x - 2$ , and the domain is all real numbers.

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QuestionFind the simplified expression of $g(x) - f(x)$ , where $f(x) = 5x + 3$ and $g(x) = x^2 - x + 1$ . Then, determine the domain of the simplified expression.

Studdy Solution

STEP 1

STEP 2

1. Perform the function operation $g(x) - f(x)$ .
2. Simplify the result of the function operation.
3. Determine the domain of the resulting function.

STEP 3

Perform the function operation $g(x) - f(x)$ .
$g(x) - f(x) = (x^2 - x + 1) - (5x + 3)$

STEP 4

Distribute the negative sign to the terms in $f(x)$ .
$g(x) - f(x) = x^2 - x + 1 - 5x - 3$

STEP 5

Combine like terms.
$g(x) - f(x) = x^2 - 6x - 2$

STEP 6

The simplified result of the function operation is:
$g(x) - f(x) = x^2 - 6x - 2$

STEP 7

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QuestionFind the simplified expression of g(x)−f(x)g(x) - f(x)g(x)−f(x), where f(x)=5x+3f(x) = 5x + 3f(x)=5x+3 and g(x)=x2−x+1g(x) = x^2 - x + 1g(x)=x2−x+1. Then, determine the domain of the simplified expression.

Studdy Solution

STEP 1

STEP 2

1. Perform the function operation g(x)−f(x)g(x) - f(x)g(x)−f(x).2. Simplify the result of the function operation.3. Determine the domain of the resulting function.

STEP 3

Perform the function operation g(x)−f(x)g(x) - f(x)g(x)−f(x).g(x)−f(x)=(x2−x+1)−(5x+3) g(x) - f(x) = (x^2 - x + 1) - (5x + 3) g(x)−f(x)=(x2−x+1)−(5x+3)

STEP 4

Distribute the negative sign to the terms in f(x)f(x)f(x).g(x)−f(x)=x2−x+1−5x−3 g(x) - f(x) = x^2 - x + 1 - 5x - 3 g(x)−f(x)=x2−x+1−5x−3

STEP 5

Combine like terms.g(x)−f(x)=x2−6x−2 g(x) - f(x) = x^2 - 6x - 2 g(x)−f(x)=x2−6x−2

STEP 6

The simplified result of the function operation is:g(x)−f(x)=x2−6x−2 g(x) - f(x) = x^2 - 6x - 2 g(x)−f(x)=x2−6x−2

STEP 7

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QuestionFind the simplified expression of $g(x) - f(x)$ , where $f(x) = 5x + 3$ and $g(x) = x^2 - x + 1$ . Then, determine the domain of the simplified expression.

1. Perform the function operation $g(x) - f(x)$ .
2. Simplify the result of the function operation.
3. Determine the domain of the resulting function.

Perform the function operation $g(x) - f(x)$ .
$g(x) - f(x) = (x^2 - x + 1) - (5x + 3)$

Distribute the negative sign to the terms in $f(x)$ .
$g(x) - f(x) = x^2 - x + 1 - 5x - 3$

Combine like terms.
$g(x) - f(x) = x^2 - 6x - 2$

The simplified result of the function operation is:
$g(x) - f(x) = x^2 - 6x - 2$