What is the domain of (x)? f(x)=-4x + 5 g(x), №17886713, 12.04.2022 04:00

Mathematics

What is the domain of (x)? f(x)=-4x + 5 g(x) = -x- 2x - 1

Step-by-step answer

P Answered by PhD

I think both domains are open, that means that it can be anything

I'm not sure though.

Mathematics

1. let f(x) = 4x-3/x-10 and g(x) = 2x-8/x-10. find (f+g)(x). assume all appropriate restrictions to the domain. a) (f+g)(x) = 6x-11/x-10 b) (f+g)(x) = 2x+5/x-10 c) (f+g)(x) = 6x+11/x-10 d) (f+g)(x) = 6x-11/2x-20 2. find the domain of the function (f/g)(x) where f(x)=x²-9 and g(x)=x²-4x+3. a) (-∞,∞) b) (-∞,1) u (1, ∞) c) (-∞,1) u (1,3) u (3, ∞) d) (-∞,-3) u (-3,-1) u (-1, ∞) 3. let f(x) = √x-2 and g(x) = √x+7. find (f*g)(x). assume all appropriate restrictions to the domain. a) (f*g)(x) = x^2+5x-14 b) (f*g)(x) = x^2+9x-14 c) (f*g)(x) = √x²+9x-14

Step-by-step answer

P Answered by Master

22

The correct answers are:

(1) Option (a) $(f+g)(x) = \frac{6x-11}{x-10}$

(2) Option (b) $(-\infty, 1) ~\bigcup ~(1, +\infty)$

(3) Option (d) $(f*g)(x) = \sqrt{x^2 + 5x - 14}$

(4) Graph is attached with the answer along with the explanation (below)!

(5) Option (b) $[3, \infty)$

Explanations:

(1) Given Data:

f(x) = $\frac{4x-3}{x-10}$

g(x) = $\frac{2x-8}{x-10}$

Required = (f+g)(x) = ?

The expression (f+g)(x) is nothing but the addition of f(x) and g(x). Therefore, in order to find (f+g)(x), we need to add both the given functions as follows:

$(f+g)(x) = \frac{4x-3}{x-10} + \frac{2x-8}{x-10}$

Now we need to simplify the above equation as follows:

$(f+g)(x) = \frac{4x-3 + 2x - 8}{x-10} \\ (f+g)(x) = \frac{6x-11}{x-10}$

Hence the correct answer is $(f+g)(x) = \frac{6x-11}{x-10}$ Option (a)

(2) Given Data:

f(x) = x^2 - 9

g(x) = x^2 - 4x + 3

Before finding the domain of the expression $(\frac{f}{g})(x)$ , we need to first evalute that expression as follows:

$(\frac{f}{g})(x) = \frac{f(x)}{g(x)} \\ Plug~in~the~values~of~f(x)~and~g(x)~in~the~above~equation.\\ (\frac{f}{g})(x) = \frac{x^2-9}{x^2-4x+3} \\ (\frac{f}{g})(x) = \frac{(x-3)(x+3)}{(x-3)(x-1)} \\ (\frac{f}{g})(x) = \frac{(x+3)}{(x-1)}$

Now we need to put the denominator equal to zero in order to know what values of x should not be in the domain of this function:

x-1 = 0

x = 1

It means that the domain of $(\frac{f}{g})(x)$ is all real numbers EXCEPT x = 1. The (closed) parentheses " ) " or "(" means that the number is not included in the domain. Therefore, we can write that the domain of $(\frac{f}{g})(x)$ is $(-\infty, 1) ~\bigcup ~(1, +\infty)$ (Option b)

(3) Given Data:

f(x) = $\sqrt{x-2}$

g(x) = $\sqrt{x+7}$

Required = (f*g)(x) = ?

The expression (f*g)(x) is nothing but the multiplication of f(x) and g(x). Therefore, in order to find (f*g)(x), we need to multiply both the given functions as follows:

$(f*g)(x) = \sqrt{x-2} * \sqrt{x+7}$

Now we need to simplify the above equation as follows:

$(f*g)(x) = \sqrt{x-2} * \sqrt{x+7} \\ (f*g)(x) = \sqrt{(x-2)(x+7)} \\ (f*g)(x) = \sqrt{x^2 + 7x -2x - 14}\\ (f*g)(x) = \sqrt{x^2 + 5x - 14}$ (Option d)

(4) Given Data:

f(x) = $\frac{1}{x}$

g(x) = $\sqrt{x}$

Required = The graph of (f-g)(x) = ?

Before plotting the graph let us evalute it first. (f-g)(x) is the subtraction of g(x) from f(x). Mathematically, we can write it as:

(f-g)(x) = $\frac{1}{x} - \sqrt{x}$

Now simplify:

$(f-g)(x) = \frac{1}{x} - \sqrt{x} \\ (f-g)(x) = \frac{1-x\sqrt{x}}{x} \\$

Look at the graph attached with this answer. As you can see, at x=0, the graph shoots up! As at x=0, the value of function approaches to infinity.

(5) Given Data:

f(x) = (x+4)^2

g(x) = 3

Required = Range of (f+g)(x) = ?

Before finding the range of (f+g)(x), we first need to write the function:

(f+g)(x) = (x+4)^2 + 3

Now that we have written the function, the next step is to find the inverse of this function in order to obtain the range.To find the inverse, swap x with y, and y with x and put (f+g)(x) = y as follows:

(f+g)(x) = y = (x+4)^2 + 3

Now swap:

x = (y+4)^2 + 3

Now solve for y:

(x-3) = (y+4)^2

Take square-root on both sides:

$\sqrt{(x-3)} = y+4$

$y = \sqrt{(x-3)} - 4$

As you know that the square root of negative numbers are the complex numbers, and in range, we do not include the complex numbers. Therefore, the values of x should be greater or equal to 3 to have the square-roots to be the real numbers. Therefore,

Range of (f+g)(x) = $[3, \infty)$ (Option b)

Note: "[" or "]" bracket is used to INCLUDE the value. It means that 3 is included in the range.

Mathematics

1. let f(x) = 4x-3/x-10 and g(x) = 2x-8/x-10. find (f+g)(x). assume all appropriate restrictions to the domain. a) (f+g)(x) = 6x-11/x-10 b) (f+g)(x) = 2x+5/x-10 c) (f+g)(x) = 6x+11/x-10 d) (f+g)(x) = 6x-11/2x-20 2. find the domain of the function (f/g)(x) where f(x)=x²-9 and g(x)=x²-4x+3. a) (-∞,∞) b) (-∞,1) u (1, ∞) c) (-∞,1) u (1,3) u (3, ∞) d) (-∞,-3) u (-3,-1) u (-1, ∞) 3. let f(x) = √x-2 and g(x) = √x+7. find (f*g)(x). assume all appropriate restrictions to the domain. a) (f*g)(x) = x^2+5x-14 b) (f*g)(x) = x^2+9x-14 c) (f*g)(x) = √x²+9x-14

Step-by-step answer

P Answered by Specialist

22

The correct answers are:

(1) Option (a) $(f+g)(x) = \frac{6x-11}{x-10}$

(2) Option (b) $(-\infty, 1) ~\bigcup ~(1, +\infty)$

(3) Option (d) $(f*g)(x) = \sqrt{x^2 + 5x - 14}$

(4) Graph is attached with the answer along with the explanation (below)!

(5) Option (b) $[3, \infty)$

Explanations:

(1) Given Data:

f(x) = $\frac{4x-3}{x-10}$

g(x) = $\frac{2x-8}{x-10}$

Required = (f+g)(x) = ?

The expression (f+g)(x) is nothing but the addition of f(x) and g(x). Therefore, in order to find (f+g)(x), we need to add both the given functions as follows:

$(f+g)(x) = \frac{4x-3}{x-10} + \frac{2x-8}{x-10}$

Now we need to simplify the above equation as follows:

$(f+g)(x) = \frac{4x-3 + 2x - 8}{x-10} \\ (f+g)(x) = \frac{6x-11}{x-10}$

Hence the correct answer is $(f+g)(x) = \frac{6x-11}{x-10}$ Option (a)

(2) Given Data:

f(x) = x^2 - 9

g(x) = x^2 - 4x + 3

Before finding the domain of the expression $(\frac{f}{g})(x)$ , we need to first evalute that expression as follows:

$(\frac{f}{g})(x) = \frac{f(x)}{g(x)} \\ Plug~in~the~values~of~f(x)~and~g(x)~in~the~above~equation.\\ (\frac{f}{g})(x) = \frac{x^2-9}{x^2-4x+3} \\ (\frac{f}{g})(x) = \frac{(x-3)(x+3)}{(x-3)(x-1)} \\ (\frac{f}{g})(x) = \frac{(x+3)}{(x-1)}$

Now we need to put the denominator equal to zero in order to know what values of x should not be in the domain of this function:

x-1 = 0

x = 1

It means that the domain of $(\frac{f}{g})(x)$ is all real numbers EXCEPT x = 1. The (closed) parentheses " ) " or "(" means that the number is not included in the domain. Therefore, we can write that the domain of $(\frac{f}{g})(x)$ is $(-\infty, 1) ~\bigcup ~(1, +\infty)$ (Option b)

(3) Given Data:

f(x) = $\sqrt{x-2}$

g(x) = $\sqrt{x+7}$

Required = (f*g)(x) = ?

The expression (f*g)(x) is nothing but the multiplication of f(x) and g(x). Therefore, in order to find (f*g)(x), we need to multiply both the given functions as follows:

$(f*g)(x) = \sqrt{x-2} * \sqrt{x+7}$

Now we need to simplify the above equation as follows:

$(f*g)(x) = \sqrt{x-2} * \sqrt{x+7} \\ (f*g)(x) = \sqrt{(x-2)(x+7)} \\ (f*g)(x) = \sqrt{x^2 + 7x -2x - 14}\\ (f*g)(x) = \sqrt{x^2 + 5x - 14}$ (Option d)

(4) Given Data:

f(x) = $\frac{1}{x}$

g(x) = $\sqrt{x}$

Required = The graph of (f-g)(x) = ?

Before plotting the graph let us evalute it first. (f-g)(x) is the subtraction of g(x) from f(x). Mathematically, we can write it as:

(f-g)(x) = $\frac{1}{x} - \sqrt{x}$

Now simplify:

$(f-g)(x) = \frac{1}{x} - \sqrt{x} \\ (f-g)(x) = \frac{1-x\sqrt{x}}{x} \\$

Look at the graph attached with this answer. As you can see, at x=0, the graph shoots up! As at x=0, the value of function approaches to infinity.

(5) Given Data:

f(x) = (x+4)^2

g(x) = 3

Required = Range of (f+g)(x) = ?

Before finding the range of (f+g)(x), we first need to write the function:

(f+g)(x) = (x+4)^2 + 3

Now that we have written the function, the next step is to find the inverse of this function in order to obtain the range.To find the inverse, swap x with y, and y with x and put (f+g)(x) = y as follows:

(f+g)(x) = y = (x+4)^2 + 3

Now swap:

x = (y+4)^2 + 3

Now solve for y:

(x-3) = (y+4)^2

Take square-root on both sides:

$\sqrt{(x-3)} = y+4$

$y = \sqrt{(x-3)} - 4$

As you know that the square root of negative numbers are the complex numbers, and in range, we do not include the complex numbers. Therefore, the values of x should be greater or equal to 3 to have the square-roots to be the real numbers. Therefore,

Range of (f+g)(x) = $[3, \infty)$ (Option b)

Note: "[" or "]" bracket is used to INCLUDE the value. It means that 3 is included in the range.

Mathematics

Let f(x) = -4x - 2 and g(x) = 5x - 6. find f*g and state its domain. 8x2 + 34x - 30; all real numbers 8x2 + 34x - 30; all real numbers except x = 1 -20x2 + 14x + 12; all real numbers -20x2 + 14x + 12; all real numbers except x = 6

Step-by-step answer

P Answered by PhD

4

Option 3. (-20x2 + 14x + 12; domain: all real numbers) is the right answer.

Step-by-step explanation:

Let f(x) = -4x - 2 and g(x) = 5x - 6

then f(x)×g(x) = (-4x-2)(5x-6)

= -(4x+2)(5x-6)

= -( 20x²-24x+10x-12)

= -( 20x²-14x-12)

f(x)×g(x) = -20x² + 14x + 12

Since we know domain: f(x) ∈ R

Similarly for g(x) domain: g(x) ∈ R

Then domain of multiplication of both the function will be domain: f(x)×g(x) ∈ R.

Mathematics

Let f(x) = -4x - 2 and g(x) = 5x - 6. find fxg and state its domain. a. 8x^2 + 34x - 30; all real numbers except x = 1 b. -20x^2 + 14x + 12; all real numbers except x = 6 c.-20x^2 + 14x + 12; all real numbers d.8x^2 + 34x - 30; all real numbers

Step-by-step answer

P Answered by PhD

1

c.-20x^2 + 14x + 12; all real numbers

Step-by-step explanation:

f(x) = -4x - 2

g(x) = 5x - 6

Therefore, -20x^2 + 14x + 12; all real numbers.

Mathematics

Let f(x) = -4x - 2 and g(x) = 5x - 6. find f*g and state its domain. 8x2 + 34x - 30; all real numbers 8x2 + 34x - 30; all real numbers except x = 1 -20x2 + 14x + 12; all real numbers -20x2 + 14x + 12; all real numbers except x = 6

Step-by-step answer

P Answered by PhD

4

Option 3. (-20x2 + 14x + 12; domain: all real numbers) is the right answer.

Step-by-step explanation:

Let f(x) = -4x - 2 and g(x) = 5x - 6

then f(x)×g(x) = (-4x-2)(5x-6)

= -(4x+2)(5x-6)

= -( 20x²-24x+10x-12)

= -( 20x²-14x-12)

f(x)×g(x) = -20x² + 14x + 12

Since we know domain: f(x) ∈ R

Similarly for g(x) domain: g(x) ∈ R

Then domain of multiplication of both the function will be domain: f(x)×g(x) ∈ R.

Mathematics

Let f(x) = -4x - 2 and g(x) = 5x - 6. find f×g and state its domain

Step-by-step answer

P Answered by PhD

see explanation

Step-by-step explanation:

f(x) × g(x)

= (4x - 2)(5x - 6) ← expand factors

= 20x² - 24x - 10x + 12 ← collect like terms

= 20x² - 34x + 12

This is a polynomial of degree 2 and is defined for all real values of x

domain : x ∈ R

Mathematics

Let f(x) = -4x - 2 and g(x) = 5x - 6. find fxg and state its domain. a. 8x^2 + 34x - 30; all real numbers except x = 1 b. -20x^2 + 14x + 12; all real numbers except x = 6 c.-20x^2 + 14x + 12; all real numbers d.8x^2 + 34x - 30; all real numbers

Step-by-step answer

P Answered by PhD

1

c.-20x^2 + 14x + 12; all real numbers

Step-by-step explanation:

f(x) = -4x - 2

g(x) = 5x - 6

Therefore, -20x^2 + 14x + 12; all real numbers.

Mathematics

Let f(x) = -4x - 2 and g(x) = 5x - 6. find f⋅g and state its domain.

Step-by-step answer

P Answered by PhD

2

-20x^2 +14x+12

The domain of f*g is all real numbers

Step-by-step explanation:

f(x) = -4x - 2

g(x) = 5x - 6.

f*g = (-4x-2) * (5x-6)

FOIL

first -4x*5x = -20x^2

outer -4x*-6 = 24x

inner -2*5x = -10x

last -2 *-6 = 12

Add them together

-20x^2 +24x-10x+12 = -20x^2 +14x+12

The domain of f is all real numbers, the domain of g is all real numbers

The domain of f*g is all real numbers

Mathematics

Find the domain of f(x) = -4x + 2 for the range {-5, 2, 6, 18).

Step-by-step answer

P Answered by PhD

Step-by-step explanation:

calculate x for this equation

f(x) = -5 : -4x +2 = - 5 so : -4x =-7 x = 7/4

f(x) = 2 : -4x +2 = 2 so : -4x =0 x = 0

f(x) =6 : -4x +2 = 6 so : -4x =4 x = -1

f(x) = 18 : -4x +2 = 18 so : -4x =16 x = - 4

the domain is : D = {7/4, 0, -1, -4}

What is the domain of
(x)?
f(x)=-4x + 5
g(x) = -x- 2x - 1

Step-by-step answer

Faq

Try asking the Studen AI a question.