Math / Algebra

QuestionFind $f+g, f-g, f g$ and $\frac{f}{g}$ . Determine the domain for each function. $f(x)=6 x-5, g(x)=x+6$ $(f+g)(x)=$ $\square$ (Simplify your answer.)

Studdy Solution

STEP 1

1. We are given two functions $f(x) = 6x - 5$ and $g(x) = x + 6$ .
2. We need to find the expressions for $f+g$ , $f-g$ , $fg$ , and $\frac{f}{g}$ .
3. We also need to determine the domain for each resulting function.

STEP 2

1. Calculate $(f+g)(x)$ and determine its domain.
2. Calculate $(f-g)(x)$ and determine its domain.
3. Calculate $(fg)(x)$ and determine its domain.
4. Calculate $\left(\frac{f}{g}\right)(x)$ and determine its domain.

STEP 3

Calculate $(f+g)(x)$ :
$(f+g)(x) = f(x) + g(x) = (6x - 5) + (x + 6)$
Simplify the expression:
$(f+g)(x) = 6x - 5 + x + 6 = 7x + 1$
The domain of $f+g$ is all real numbers, $\mathbb{R}$ , because both $f(x)$ and $g(x)$ are linear functions with no restrictions.

STEP 4

Calculate $(f-g)(x)$ :
$(f-g)(x) = f(x) - g(x) = (6x - 5) - (x + 6)$
Simplify the expression:
$(f-g)(x) = 6x - 5 - x - 6 = 5x - 11$
The domain of $f-g$ is all real numbers, $\mathbb{R}$ , because both $f(x)$ and $g(x)$ are linear functions with no restrictions.

STEP 5

Calculate $(fg)(x)$ :
$(fg)(x) = f(x) \cdot g(x) = (6x - 5)(x + 6)$
Expand the expression:
$(fg)(x) = 6x^2 + 36x - 5x - 30 = 6x^2 + 31x - 30$
The domain of $fg$ is all real numbers, $\mathbb{R}$ , because polynomial functions are defined for all real numbers.

STEP 6

Calculate $\left(\frac{f}{g}\right)(x)$ :
$\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{6x - 5}{x + 6}$
The domain of $\frac{f}{g}$ is all real numbers except where the denominator is zero. Set the denominator equal to zero and solve for $x$ :
$x + 6 = 0$ $x = -6$
Thus, the domain of $\frac{f}{g}$ is all real numbers except $x = -6$ , or $\mathbb{R} \setminus \{-6\}$ .

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QuestionFind $f+g, f-g, f g$ and $\frac{f}{g}$ . Determine the domain for each function. $f(x)=6 x-5, g(x)=x+6$ $(f+g)(x)=$ $\square$ (Simplify your answer.)

Studdy Solution

STEP 1

1. We are given two functions $f(x) = 6x - 5$ and $g(x) = x + 6$ .
2. We need to find the expressions for $f+g$ , $f-g$ , $fg$ , and $\frac{f}{g}$ .
3. We also need to determine the domain for each resulting function.

STEP 2

1. Calculate $(f+g)(x)$ and determine its domain.
2. Calculate $(f-g)(x)$ and determine its domain.
3. Calculate $(fg)(x)$ and determine its domain.
4. Calculate $\left(\frac{f}{g}\right)(x)$ and determine its domain.

STEP 3

Calculate $(f+g)(x)$ :
$(f+g)(x) = f(x) + g(x) = (6x - 5) + (x + 6)$
Simplify the expression:
$(f+g)(x) = 6x - 5 + x + 6 = 7x + 1$
The domain of $f+g$ is all real numbers, $\mathbb{R}$ , because both $f(x)$ and $g(x)$ are linear functions with no restrictions.

STEP 4

Calculate $(f-g)(x)$ :
$(f-g)(x) = f(x) - g(x) = (6x - 5) - (x + 6)$
Simplify the expression:
$(f-g)(x) = 6x - 5 - x - 6 = 5x - 11$
The domain of $f-g$ is all real numbers, $\mathbb{R}$ , because both $f(x)$ and $g(x)$ are linear functions with no restrictions.

STEP 5

Calculate $(fg)(x)$ :
$(fg)(x) = f(x) \cdot g(x) = (6x - 5)(x + 6)$
Expand the expression:
$(fg)(x) = 6x^2 + 36x - 5x - 30 = 6x^2 + 31x - 30$
The domain of $fg$ is all real numbers, $\mathbb{R}$ , because polynomial functions are defined for all real numbers.

STEP 6

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QuestionFind f+g,f−g,fgf+g, f-g, f gf+g,f−g,fg and fg\frac{f}{g}gf​. Determine the domain for each function. f(x)=6x−5,g(x)=x+6f(x)=6 x-5, g(x)=x+6f(x)=6x−5,g(x)=x+6 (f+g)(x)=(f+g)(x)=(f+g)(x)= □\square□ (Simplify your answer.)

Studdy Solution

STEP 1

1. We are given two functions f(x)=6x−5 f(x) = 6x - 5 f(x)=6x−5 and g(x)=x+6 g(x) = x + 6 g(x)=x+6.2. We need to find the expressions for f+g f+g f+g, f−g f-g f−g, fg fg fg, and fg \frac{f}{g} gf​.3. We also need to determine the domain for each resulting function.

STEP 2

1. Calculate (f+g)(x) (f+g)(x) (f+g)(x) and determine its domain.2. Calculate (f−g)(x) (f-g)(x) (f−g)(x) and determine its domain.3. Calculate (fg)(x) (fg)(x) (fg)(x) and determine its domain.4. Calculate (fg)(x) \left(\frac{f}{g}\right)(x) (gf​)(x) and determine its domain.

STEP 3

STEP 4

STEP 5

STEP 6

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QuestionFind $f+g, f-g, f g$ and $\frac{f}{g}$ . Determine the domain for each function. $f(x)=6 x-5, g(x)=x+6$ $(f+g)(x)=$ $\square$ (Simplify your answer.)

1. We are given two functions $f(x) = 6x - 5$ and $g(x) = x + 6$ .
2. We need to find the expressions for $f+g$ , $f-g$ , $fg$ , and $\frac{f}{g}$ .
3. We also need to determine the domain for each resulting function.

1. Calculate $(f+g)(x)$ and determine its domain.
2. Calculate $(f-g)(x)$ and determine its domain.
3. Calculate $(fg)(x)$ and determine its domain.
4. Calculate $\left(\frac{f}{g}\right)(x)$ and determine its domain.