Question

How do you find the range given $x={{t}^{2}}-4$ and $y=\dfrac{t}{2}$ for $-2\le t\le 3$ ?

Answer 1

Hint: In order to find the range of the given function, we must first solve the inequality of the variable $t$. By our prior knowledge of the domain and range of a function, we see that the range of the function is given by the variable $y$. We shall then find the value of variable $y$ which represents the range of the function by substituting the appropriate values of $t$ obtained before.

Complete step by step answer:
If we have a function, let say $f$ , and if we give it a valid input of variable $x$, then this function is going to map that to an output which we would call $f\left( x \right)$. This output is also represented as the variable $y$.

Thus, the range of the function is the set of all possible outputs that the function can produce.
Since, we know that $y$ gives the range of the function, therefore, we must find an interval of values of$y$.
Given that, $y=\dfrac{t}{2}$, thus we shall find the value of $\dfrac{t}{2}$ first as it is equal to $y$.
Also, given that $-2\le t\le 3$, so we will divide this entire inequality by $2$.
$\Rightarrow \dfrac{-2}{2}\le \dfrac{t}{2}\le \dfrac{3}{2}$
$\Rightarrow -1\le \dfrac{t}{2}\le \dfrac{3}{2}$
This implies that $-1\le y\le \dfrac{3}{2}$ because$y=\dfrac{t}{2}$.
Therefore, the range of the given function is $\left[ -1,\dfrac{3}{2} \right]$.

Note: A domain is the set of all of the inputs over which the function is defined. If we input a value $x$ from this domain then the function will output another value, $f\left( x \right)$. However, if we put a value $x$ which is not in the domain then the function would not be able to give a definite value.