Consider the function f(x)=-2/x+4. Determine the domain and range

Question

MHF4U Rational Fuction

Answer 1

Alright, this may be a little more in depth than I would like it to be. If you have any questions, please ask :)

Let's review:

The Domain of the function, in this case, is the set of all inputs "x" that allow the function to exist.

The Range of the function, in this case, is the set of all outputs "f(x)" that exist

I am unsure if this function is f(x) = (-2/x)+4 or f(x) = -2/(x+4) so I will do an example for both.

Take f(x) = (-2/x)+4

First, let's ask ourselves if there is a value for "x" at which the function does not exist

In this case, when x = 0 the function does not exist because you cannot divide by 0.

Since there are no other values of x where the function is undefined, we can say that the range of x is all real numbers, where x does not equal 0.

Next, lets plug in numbers for x, where x does not equal 0, to see how the output "f(x)" responds.

x = 1, f(x) = 2

x = 10, f(x) = 3.8

x = 100, f(x) = 3.98

Now, we can see that as x approaches positive infinity, -2/x goes to zero, leaving 4 as an upper limit

Let's try negative numbers:

x = -1, f(x) = 6

x = -10, f(x) = 4.2

x = -100, f(x) = 4.02

As x approaches negative infinity, -2/x approaches zero, leaving 4 as a lower limit. This means that f(x) can get as close to 4 as it wants but can never equal 4.

Now we gather, f(x) can equal all numbers except 4.

Putting f(x) and x together, we get the

Domain: All Real Numbers where x does not equal 0

Range: All Real Numbers where f(x) does not equal 4

Next Example: f(x) = -2/(x+4)

Following the previous instructions, we get x cannot equal -4 because -4+4 = 0 and we cannot divide anything by zero.

So x can be all real numbers such that x does not equal -4.

Next we plug in values for x to see how f(x) responds:

x = 1, f(x) = -2/5

x = 100, f(x) = -2/104

x = 1000, f(x) = -2/1004

so as x goes to positive infinity, f(x) goes to zero. This means that f(x) can get as close to zero as it wants without ever being equal to zero.

x = -1, f(x) = -2/3

x = -10, f(x) = 2/6

x = -100, f(x) = 2/96

So as x goes to negative infinity, f(x) can get as close as it wants to 0, without ever being zero.

We can now state the Domain and Range:

Domain: all real numbers where x does not equal -4

Range: all real numbers where f(x) does not equal 0

Hope this helps!