Question

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

Answer 1

Ans(a):

Given function is $f(x)=\frac{3x-1}{x+4}$

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

$g(x)=\frac{3x-1}{x+4}+4$

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

$8=\frac{3x-1}{x+4}+4$

$8-4=\frac{3x-1}{x+4}$

$4=\frac{3x-1}{x+4}$

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

4x+16=3x-1

4x-3x=-1-16

x=-17

Hence final answer is x=-17

Answer 2

Answer:

Part a) $x=-4$

Part b) $g(x)=\frac{3x-1}{x+4}+4$

Part c) $x=-17$

Step-by-step explanation:

we have

$f(x)=\frac{3x-1}{x+4}$

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

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

$g(x)=\frac{3x-1}{x+4}+4$

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

$8=\frac{3x-1}{x+4}+4$

$8-4=\frac{3x-1}{x+4}$

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

$4x+16=3x-1$

$4x-3x=-16-1$

$x=-17$