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Domain and Range of Rational Functions

TopIn a rational function, the domain of function f (t) is basically the set of values, where by putting the values of ‘t’ in function, f(t) is always real. The range of function f (t) is basically a set of values, where the function f(t) takes, when the variable ‘t’ takes values in the domain. To understand domain and range of rational functions, we take an example of rational function f(t).

Let f(t) = $\frac{t + 3}{t^2 - 5t - 66}$

Now, to find the domain, we first factor the denominator of the function f (t). The denominator of function f(t) is t2 - 5t - 66.
To factor this, we divide -5t, such that addition becomes -5t and multiplication becomes -66. So,
t2 - 11t + 5t - 66 = 0
t(t - 11) + 5(t - 11) = 0
(t - 11) (t + 5) = 0

So, the term t2 - 5t - 66 contains two factors (t - 11) and (t + 5). Here, the domain of function f (t) is all values of ‘t’ which can be substituted and answer is always defined.

So, denominator = (t - 12) (t + 2). So, the domain of function f (t) = $(- \infty, -2) U (-2, 11) U (11, \infty)$.

Now, we have to find out the range.

So, f (t) = $\frac{t + 3}{t^2 - 5t - 66}$

y = $\frac{t + 3}{t^2 - 5t - 66}$

After that, we multiply both the sides by the Least Common Denominator,
y(t2 - 5t - 66) = (t + 3)
yt2 - 5yt - t - 66y - 3 = 0
yt2 - (5y + 1) t - 66y - 3 = 0
yt2 - (5y + 1) t - 3(22y + 1) = 0
Resulting equation is looking like Quadratic Equation and the range of all real numbers is $(-\infty, \infty)$. So, the range of the function f (t) is also $(-\infty, \infty)$.