Finding the range of a function

[jonnburton]

Hi Everyone,

Could anyone tell me how to find the range of this function:

\(\displaystyle f(x) = \frac{2x+5}{x-3}, x\geqslant 5\)

It is clear to me that the domain includes all real numbers except for x =3 but I am not sure about the range. The book I am using says the range includes all real numbers except for 2 but I am not really sure at how this is obtained.

 

[Deleted member 4993]

Hi Everyone,

Could anyone tell me how to find the range of this function:

\(\displaystyle f(x) = \frac{2x+5}{x-3}, x\geqslant 5\)

It is clear to me that the domain includes all real numbers except for x =3 but I am not sure about the range. The book I am using says the range includes all real numbers except for 2 but I am not really sure at how this is obtained.

Hint:

f(x) = (2x+5)/(x-3) = 2 + \(\displaystyle \frac{11}{x-3}\)

In defining range - you need to test for horizontal asymptotes.

 

[jonnburton]

Ah, thank you Subhotosh!

The output from this function will always be 2 + a fraction, therefore it can never equal 2.

 

[Deleted member 4993]

Hi Everyone,

Could anyone tell me how to find the range of this function:

\(\displaystyle f(x) = \frac{2x+5}{x-3}\), \(\displaystyle x\geqslant 5\)

It is clear to me that the domain includes all real numbers except for x =3 but I am not sure about the range. The book I am using says the range includes all real numbers except for 2 but I am not really sure at how this is obtained.

If you posted the problem correctly - then the domain of the function has been defined to be x≥5.

In that case the range is [7.5,2) or 7.5 ≥ y > 2.

 

[jonnburton]

Thanks Subhotosh. I understand that. I do need to practice this type of problem more as I'm finding them quite tricky.

 

[HallsofIvy]

Hi Everyone,

Could anyone tell me how to find the range of this function:

\(\displaystyle f(x) = \frac{2x+5}{x-3}, x\geqslant 5\)

It is clear to me that the domain includes all real numbers except for x =3 but I am not sure about the range. The book I am using says the range includes all real numbers except for 2 but I am not really sure at how this is obtained.

As you have seen another way to do this is to write the function as \(\displaystyle y= \frac{2x+ 5}{x- 3}\). Now solve the equation for x as a function of y. Multiplying both sides by x- 3, y(x- 3)= xy- 3y= 2x+ 5. Add 3y and subtract 2x from both sides: xy- 2x= x(y- 2)= 3y+ 5. Finally, divide both sides by y- 2: \(\displaystyle x= \frac{3y+ 5}{y- 2}\). Since we cannot divide by 0, y cannot be 2.