# Range of function

• Jun 6th 2009, 02:34 AM
champrock
Range of function
Hi

What will be the range of the function f(x) = 1-1/x .

1. The domain is given as [0, infinity)

2. The domain is given as [0, infinity]

I am primarily confused if "1" will be included or not? The limit of the function tends to 1 but it will never actually reach 1. So, will the range be [-infinity,1) or [-infinity,1] or (-infinity,1) or (-infinity,1]

thanks
• Jun 6th 2009, 02:58 AM
Prove It
Quote:

Originally Posted by champrock
Hi

What will be the range of the function f(x) = 1-1/x .

1. The domain is given as [0, infinity)

2. The domain is given as [0, infinity]

I am primarily confused if "1" will be included or not? The limit of the function tends to 1 but it will never actually reach 1. So, will the range be [-infinity,1) or [-infinity,1] or (-infinity,1) or (-infinity,1]

thanks

Hint: Find the range of the function $\displaystyle f(x) = \frac{1}{x}$ in the domain $\displaystyle x \geq 0$.

The multiplication by $\displaystyle -1$ gives a reflection in the x-axis, and the addition of 1 is a translation of 1 unit vertically upward.
• Jun 6th 2009, 03:10 AM
Amer
I write the answer but when I see the member above me give you a hint I edit it

note you can't write the interval closed from the direction of infinity
like this

(0,infinity]

the correct like this

(0,infinity)
• Jun 6th 2009, 04:20 AM
champrock
Quote:

Originally Posted by Amer
I write the answer but when I see the member above me give you a hint I edit it

note you can't write the interval closed from the direction of infinity
like this

(0,infinity]

the correct like this

(0,infinity)

hmm ok . (0,infinity) is because infinity cant be included in the set right? Its not any proper number.

@Prove It : I found the range. Its coming out to be -infinity to 1. But I am not sure if 1 is actually included or not because
Lim 1/x (x tends to infinity) ---> 0
so 1-1/x = 1-0 = 1
but
But 1/(some very large number ) ----> some very tiny number like 0.000000000045
So, 1-1/x = 1-some very tiny number. = 0.99999999999999434

SO, the range is (-infinity,1) or (-infinity,1] ?
• Jun 6th 2009, 04:22 AM
Amer
the first one (-infinity,1)

thanks to prove it
• Jun 6th 2009, 04:40 AM
Prove It
Quote:

Originally Posted by champrock
hmm ok . (0,infinity) is because infinity cant be included in the set right? Its not any proper number.

@Prove It : I found the range. Its coming out to be -infinity to 1. But I am not sure if 1 is actually included or not because
Lim 1/x (x tends to infinity) ---> 0
so 1-1/x = 1-0 = 1
but
But 1/(some very large number ) ----> some very tiny number like 0.000000000045
So, 1-1/x = 1-some very tiny number. = 0.99999999999999434

SO, the range is (-infinity,1) or (-infinity,1] ?

Let's put it this way...

In the domain $\displaystyle x > 0$, (since $\displaystyle x \neq 0$ as pointed out earlier), the range of the function $\displaystyle \frac{1}{x}$ is $\displaystyle y > 0$, or in interval notation, $\displaystyle (0, \infty)$.

Notice that as $\displaystyle x \to \infty, y \to 0$. BUT since we can never reach $\displaystyle \infty$, $\displaystyle y$ will never reach $\displaystyle 0$.

Now the reflection in the $\displaystyle x$ axis means that the range of $\displaystyle -\frac{1}{x}$ becomes $\displaystyle y < 0$, or to use interval notation, $\displaystyle (-\infty, 0)$.

The translation of 1 unit vertically upward means that the range of $\displaystyle 1 - \frac{1}{x}$ is $\displaystyle y < 1$, or $\displaystyle (-\infty, 1)$ in interval notation.

So to answer your question, you do NOT include the 1 in the range.