Find the Domain and Range
Find the Domain and Range - How it Works - Video
Example 1
Example 1:
Here we f(x) = x2 - 3.
Remember the domain is the set of all the input values a.k.a. the x-values.
Remember the range is the set of all the output values a.k.a. the y-values.
Let's look at our graph. Since we have a quadratic, we know that we have a u-shape graph. Now our domain is all the horizontal numbers or x-values. The graph will continue to go left and right albeit rather slowly compared the y-values. It is still going left and right.
We don't have a denominator or logs, square roots, or anything to hinder what number we input into the function. So, our domain is all real numbers of x. We can write that shorter with symbols as x ∈ ℝ (we read that as x is the element of all real numbers). Or we can use interval notation, (-∞, +∞).
Our range is all the vertical numbers or y-values. The graph will continue to go up and up forever and ever, but it stops going downward the the point (0, -3) because (0, -3) is the vertex of our quadratic function. Since we have a minimum, our graph can't go below that value. So our range is y ≥ -3 or using interval notation we have [-3, +∞).
Example 2
Example 2:
Here we f(x) = sqrt(x + 2).
Remember the domain is the set of all the input values a.k.a. the x-values.
Remember the range is the set of all the output values a.k.a. the y-values.
Let's look at our graph. Now our domain is all the horizontal numbers or x-values. The graph will continue to go right for forever and ever, but it stops at -2 when we move left. How did we get that value? Well, we look inside the radical and we have x + 2. Our graph translate two units to the left because of the +2. The minimum x-value for square root function, sqrt(x), without transformations is x = 0. So we set x + 2 = 0 and we get x = -2. So our domain is x ≥ -2 or using interval notation we have [-2, +∞).
This time we did have a square root so we hinder our input values because of the limitations of the function.
Our range is all the vertical numbers or y-values. The graph will continue to go up and up forever and ever, but it stops going downward the the point (-2, 0). Our graph can't go below that value. So our range is y ≥ 0 or using interval notation we have [0, +∞).
Example 3
Example 3:
Here we f(x) = 4/x.
Remember the domain is the set of all the input values a.k.a. the x-values.
Remember the range is the set of all the output values a.k.a. the y-values.
Let's look at our graph. We have a reciprocal function. Now our domain is all the horizontal numbers or x-values. The graph will continue to go left and right except at one value. We have one vertical asymptote.
We have a denominator so have to be careful with our domain since we can't have 0 in the denominator. Only one value gives us 0 in the denominator and it is x = 0. So, our domain is all real numbers of x except x cannot equal 0. We can write that shorter with symbols as x ∈ ℝ, x ≠ 0 (we read that as x is the element of all real numbers except x cannot equal 0). We can also write that is x > 0 or x < 0, which interval notation is (-∞, 0) U (0, +∞).
Our range is all the vertical numbers or y-values. The graph will continue to go up and up forever and ever, but there is one value that our function cannot output and that y = 0 since we have a horizontal asymptote at y = 0. So, our range is all real numbers of y except y cannot equal 0. We can write that shorter with symbols as y ∈ ℝ, y ≠ 0 (we read that as y is the element of all real numbers except y cannot equal 0). We can also write that is y > 0 or xy< 0, which interval notation is (-∞, 0) U (0, +∞).