Question

Consider the function f(x) = 3x-1/x+4.

(a) At which value of x will the function not have a solution? Explain your answer.

(b) If g(x) is a vertical shift of 4 units of f(x), write the function of g(x). How does the graph of g(x) compare to the graph of f(x)? Explain the function you wrote.

(c) What is the value of x when g(x) = 8? Show your work.

Answer 1

Part a)
`x=-4`

Part b)
`g(x)=(3x-1)/(x+4)+4`

Part c)
`x=-17`

Explanation:

we have

`f(x)=(3x-1)/(x+4)`

Part a) At which value of x will the function not have a solution?

we know that

In the function f(x) the denominator can not be zero

therefore

To find the value of x at which the function not have solution, equate the denominator to zero

`x+4=0`

solve for x

`x=-4`

Part b) If g(x) is a vertical shift of
`4` units of f(x), write the function of g(x). How does the graph of g(x) compare to the graph of f(x)?

In this problem the rule of the translation f(x)-----> g(x) is

`(x,y)------> (x,y+4)`

That means-----> the translation is
`4` units up

The function g(x) is equal to

`g(x)=f(x)+4`

`g(x)=(3x-1)/(x+4)+4`

Part c) What is the value of x when
`g(x)= 8`?

Substitute the value of
`g(x)= 8` in the equation and solve for x

`8=(3x-1)/(x+4)+4`

`8-4=(3x-1)/(x+4)`

`4(x+4)=3x-1`

`4x+16=3x-1`

`4x-3x=-16-1`

`x=-17`

Answer 2

Ans(a):

Given function is
`f(x)=(3x-1)/(x+4)`

we know that any rational function is not defined when denominator is 0 so that means denominator x+4 can't be 0

so let's solve

x+4≠0 for x

x≠0-4

x≠-4

Hence at x=4, function can't have solution.

Ans(b):

We know that vertical shift occurs when we add something on the right side of function so vertical shift by 4 units means add 4 to f(x)

so we get:

g(x)=f(x)+4

`g(x)=(3x-1)/(x+4)+4`

We may simplify this equation but that is not compulsory.

Comparision:

Graph of g(x) will be just 4 unit upward than graph of f(x).

Ans(c):

To find value of x when g(x)=8, just plug g(x)=8 in previous equation

`8=(3x-1)/(x+4)+4`

`8-4=(3x-1)/(x+4)`

`4=(3x-1)/(x+4)`

`4(x+4)=(3x-1)`

4x+16=3x-1

4x-3x=-1-16

x=-17

Hence final answer is x=-17