find the domain and range of the function f(x)=x^2-9/4x^2+x

You have

(x-3)(x+3) / x(4x+1)

The domain is all real numbers except where the denominator is zero. So, that would be all reals except 0 and -1/4

There is a horizontal asymptote at y = 1/4.
There are vertical asymptotes at x = -1/4 and x=0.

For x between -1/4 and 0, y > 0
As x gets big, y lies below the asymptote.

So, the range outside (-1/4,0) is all reals less than 1/4.

y' = (x^2+72x+9)/(4x^2+1)^2
y' = 0 when x = -.125
y(-.125) = 143.75

So, the range is all reals
< 1/4 or >= 143.75

To find the domain and range of a function, we need to consider any restrictions or limitations on the values of x and the possible values of f(x).

First, let's find the domain of the function f(x). The domain represents all possible x-values for which the function is defined. In this case, the function f(x) is defined for all real numbers except the values that make the denominator zero since division by zero is undefined. Therefore, the denominator x^2 + x cannot be equal to zero.

To find the values that make the denominator zero, we set x^2 + x = 0 and solve for x.

x^2 + x = 0
x(x + 1) = 0

From here, we have two possible solutions:
1) x = 0
2) x + 1 = 0 --> x = -1

Therefore, the values x = 0 and x = -1 are not in the domain of the function f(x).

The domain of the function f(x) is all real numbers except x = 0 and x = -1.

Now let's find the range of the function f(x). The range represents all possible y-values (or f(x)-values) that the function can take.

We can start by simplifying the function:

f(x) = (x^2 - 9) / (4x^2 + x)

The range of a rational function is determined by the behavior of the function as x approaches positive and negative infinity.

As x approaches positive or negative infinity, the highest power terms dominate the expression. In this case, for large positive or negative values of x, the x^2 term and the 4x^2 term in the numerator and denominator respectively are the dominant terms.

The leading terms are both positive, and as x approaches positive or negative infinity, both the numerator and the denominator become positive. Therefore, the function f(x) approaches positive infinity as x approaches both positive and negative infinity.

So, the range of the function f(x) is all real numbers greater than or equal to 0.

To summarize:
- Domain: All real numbers except x = 0 and x = -1.
- Range: All real numbers greater than or equal to 0.