Question

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
d) (f*g)(x) = √x²+5x-14
4. Determine the graph of (f-g)(x) when f(x) = 1/x and g(x) = √x.
5. Find the range of (f+g)(x) when f(x)=(x+4)² and g(x)=3.
a) (-∞,∞)
b) [3, ∞)
c) (-∞, -3)
d) [-3, ∞)

Answer 1

Part 1.

Given that
`f(x) = (4x-3)/(x-10)` and
`g(x) = (2x-8)/(x-10)`

Then,
`(f+g)(x)=(4x-3)/(x-10)+(2x-8)/(x-10)=(4x-3+2x-8)/(x-10)=(6x-11)/(x-10)`.

Therefore, the correct anser is option a



Part 2.

Given f(x)=x²-9 and g(x)=x²-4x+3.

Then,
`\left((f)/(g)\right)(x)= (x^2-9)/(x^2-4x+3) = ((x-3)(x+3))/((x-1)(x-3)) =(x+3)/(x-1)`

The domain of
`\left((f)/(g)\right)(x)` is all real numbers except for the value of x for which the denominator is 0.

i.e. x - 1 = 0 implies that x = 1.

Therefore, the domain of
`\left((f)/(g)\right)(x) is (-∞,1) U (1, ∞)` [option b]



Part 3.

Given that
`f(x) = √(x-2)` and
`g(x) = √(x+7)`.

Then

`(f*g)(x)=√(x-2)*√(x+7) \\ \\ =√((x-2)(x+7))=\sqrt{x^2+5x-14`
[option d]



Part 4.
Given that
`f(x) = (1)/(x)` and
`g(x) = √(x)`.

Then,
`(f-g)(x)=(1)/(x)-√(x)`

The graph of
`(f-g)(x)=(1)/(x)-√(x)` is attached.



Part 5.

Given that
`f(x)=(x+4)^2` and
`g(x)=3`.

Then,
`(f+g)(x)=(x+4)^2+3`.

The vertex of (f + g)(x) is (-4, 3),
The range of
`(f+g)(x)=(x+4)^2+3` is all real numbers greater than or equal to 3.

Therefore, the reange of (f + g)(x) is [3, ∞) [option b]

Answer 2

The correct answers are:

(1) Option (a)
`(f+g)(x) = (6x-11)/(x-10)`

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

(3) Option (d)
`(f*g)(x) = √(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) =
`(4x-3)/(x-10)`

g(x) =
`(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) = (4x-3)/(x-10) + (2x-8)/(x-10)`

Now we need to simplify the above equation as follows:

`image`

Hence the correct answer is
`(f+g)(x) = (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
`((f)/(g))(x)`, we need to first evalute that expression as follows:

`image`

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
`((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
`((f)/(g))(x)` is
`(-\infty, 1) ~\bigcup ~(1, +\infty)` (Option b)

(3) Given Data:

f(x) =
`√(x-2)`

g(x) =
`√(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) = √(x-2) * √(x+7)`

Now we need to simplify the above equation as follows:

`image` (Option d)

(4) Given Data:

f(x) =
`(1)/(x)`

g(x) =
`√(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) =
`(1)/(x) - √(x)`

Now simplify:

`image`

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:

`√((x-3)) = y+4`

`y = √((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.