Domain & Range?

Question

What is the domain and Range of this equation:

y= (4x) / (x^2 + x - 12)

Also, how do you find this out?

Answer 1

y= (4x) / (x^2 + x - 12)
y = 4x/[(x+4)(x-3)]
When x =-4 or x = +3 the denominator is 0 so y is discontinuous at those two points.
So the domain is all real values of x except x = -4 and x = 3.
The range is all real numbers from y= - infinity to y = + infinity.

Answer 2

In general, the domain is going to be real numbers, with certain limits. For example, when the denominator of a fraction = 0, the function is undefined.

So, where is x^2 + x - 12 = 0?

This factors to (x + 4)(x - 3) or x = -4 and x = 3.

So this function has a domain of (-inf,-4), (-4, 3), (3, inf).
Replace inf with the infinity sign, the sideways 8.

Now, let's look at what happens in each section of the domain.

I use Excel for this. I put a value of -10 in cell A1. I put =(4*A1)/(A1^2 + A1 - 12) in B1.

In A2 I put = A1+0.1. I copy this formula in column A until I get a value I like, like 10. I copy the formula in B1 all the way down in column B.

From this I see that to the left of -4, the function starts out at negative infinity and goes closer and closer to zero without ever going positive. At x = -6, y = -1.333_. At x = -9, y = -0.6.

Between x = -4 and x = 3, the equation starts out very positive (f(-3.9) = 22.6, gets close to 0 quickly, f(-2) = 0.8, drifts slowly down to f(0) = 0, and then goes negative f(2) = -1.333, f(2.9) = -16.8.

On the other side of 3, f(x) starts out positive f(3.1) = 17.5. It quickly becomes asymptotic to the x-axis. f(10) = .4, f(1000) = 0.003.

So since all values are covered, the range is all real numbers.

Answer 3

Domain:

The given function is y = (4x) / ( x ^ 2 + x -12 )

set the denominator expression is equal to zero.
x^2+x - 12 = 0
( x-3)(x+4) = 0
x = 3 , - 4
So, the domain is all the real numbers except the values
x = 3 , - 4

Range:

To find the range of the given function we need to draw the graph.
In the graph y takes the maximum value 3.8(approximately)
So, y takes all the values less than or equal to 3.8.
Therefore, the range is (-infinity, +3.8 ].

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