What is image of function f(x)=-2x+1 if given domain [-2,1] and range R?
The problem provides the information that the domain of the function is the interval `[-2,1]` , hence, since the domain of the function consists of all `x` values, you need to solve the inequality `-2 <= x <= 1` but first, you need to replace `y` for `f(x)` in the equation of the given function, such that:
`y = -2x + 1`
You need to write `x` in terms of `y` , hence, you need to isolate the term that contains `x` to the left side, such that:
`2x = 1 - y => x = (1 - y)/2`
Replacing `(1 - y)/2` for `x` in the inequality you need to solve, yields:
`-2 <= (1 - y)/2 <= 1 => -2*2 <= 1 - y <= 2*1`
`-4 <= 1 - y <= 2 => -4 - 1 <= 1 - 1 - y <= 2 - 1`
`-5 <= -y <= 1`
You need to multiplicate by -1, hence, you need to change the direction of inequality, such that:
`-1 <= y <= 5`
Hence, evaluating the image of the given function, under the given conditions, yields `y in [-1,5].`
Given function f(x) is continous in in closed interval [-2,1]. Therefore it will attain its bounds.
Its bounds are
Thus function f will all values in [-1,5].
Thus image of function f is x , `x in [-1,5] sub RR` .
Thus image is all points lies on the straight line in [-1,5] on the above graph.