Question

How do you solve $ y = \dfrac{1}{4}x - 1 $ if the domain is $ \left\{ { - 4, - 2,0,2,4} \right\} $ ?

Answer 1

Hint: In order to find the value of $ y = \dfrac{1}{4}x - 1 $ for the domain is $ \left\{ { - 4, - 2,0,2,4} \right\} $ , we just need to substitute each value of domain in the given function, solve it and get the resultant which would be called as range or values of the domain.

Complete step by step solution:
We are given with a function $ y = \dfrac{1}{4}x - 1 $ , whose domains are $ \left\{ { - 4, - 2,0,2,4} \right\} $ .
We need to find the range of the given function, for that substitute the domain value in the function and find the range or values of $ y $ in domain for that.
Let’s take our first domain value that is $ \left\{ { - 4} \right\} $ , substitute it in place of $ x $ , solve it and get the value:
 $ y = \dfrac{1}{4}x - 1 = \dfrac{1}{4} \times - 4 - 1 = - 1 - 1 = - 2 $
Similarly, take the second value of the domain that is $ \left\{ { - 2} \right\} $ and repeat the steps:
 $ y = \dfrac{1}{4}x - 1 = \dfrac{1}{4} \times - 2 - 1 = \dfrac{{ - 1}}{2} - 1 = \dfrac{{ - 3}}{2} $
Solving the other domains with the same steps we get:
For $ \left\{ 0 \right\} $ :
 $ y = \dfrac{1}{4}x - 1 = \dfrac{1}{4} \times 0 - 1 = 0 - 1 = - 1 $
For $ \left\{ 2 \right\} $ :
 $ y = \dfrac{1}{4}x - 1 = \dfrac{1}{4} \times 2 - 1 = \dfrac{1}{2} - 1 = \dfrac{{ - 1}}{2} $
And for the last one that is $ \left\{ 4 \right\} $ :
 $ y = \dfrac{1}{4}x - 1 = \dfrac{1}{4} \times 4 - 1 = 1 - 1 = 0 $
Therefore, the values for the function $ y = \dfrac{1}{4}x - 1 $ if the domain is $ \left\{ { - 4, - 2,0,2,4} \right\} $ is $ \left\{ { - 2,\dfrac{{ - 3}}{2}, - 1,\dfrac{{ - 1}}{2},0} \right\} $ .
So, the correct answer is “ $ \left\{ { - 2,\dfrac{{ - 3}}{2}, - 1,\dfrac{{ - 1}}{2},0} \right\} $ ”.

Note: For any kind of a given function, just look out for the domain first, identify the restrictions for the domain if there any. If the denominator there in the function, set the denominator equal to zero and solve for x.
Write the domain in the interval form, making sure to exclude any restricted value.
There can be restrictions in the ranges also, look out for the required range before coming to the conclusion.
If we need to find the domain, always ensure to write it in the interval form.