I'm ready to give up

Question

Find the range of the function y=3/4x+17 when the domain is (-12,-4,8)

Answer 1

I encourage you to keep trying. It will come to you if you stick with it.

The notation (-12, -4.8) is called interval notation. It is worthy of noting: (a,b) means to exclude the endpoints a and b: think a < x < b. In this case the interval is called an open interval. When endpoints are included the notation is [a,b] and this is called a closed interval. Think a ≤ x ≤ b. Finally, note that half-open intervals (a,b] or [a,b) are also possible, where one endpoint is included and one is not.

A couple definitions, assuming conventional y = f(x) notation:

1. The domain of a function is the set of possible input values (x) to the function. If the domain is not explicitly given, then the domain is the set of values of x for which the function is defined. A few examples: (1) y=f(x) = 1/x --- the domain is all real x values where x ≠ 0. In interval notation one would write this (-∞, 0) U (0,∞). (2) y=f(x)=1/(x-2) --- here the domain is all real x where x ≠ 2. In interval notation, one would write (-∞, 2) U (2, ∞).
2. The range of a function is the set of possible output values corresponding to the input domain. You can think of this as what values does y take on when x is varied from one end of its domain to the other. The range can be specified in interval notation as well, using the same rules covered above.

For this problem, you have y = (3/4)x + 17 where the domain of x is (-12, -4.8)

So now we know the domain of x is -12 < x < -4.8 (not ≤, since it is an open interval)

What is the range of y? We will plug in x=-12 and x=-4.8 keeping in mind that these values of x are not really in the domain but mark the boundaries of it:

At x=-12, y = (3/4)(-12) + 17 = -9 + 17 = 8

At x=-4.8, y = (3/4)(-4.8) + 17 = -3.6 + 17 = 13.4

You can see now that the range of the function (y) is 8 < y < 13.4, in interval notation the range can be expressed as (8, 13.4)

You can graph the function to see the shape of the function and get a feel for the domain of x and range of y. On the graph, at the points° (-12,8) and (-4.8,13.4) you'd put small hollow circles ο to indicate "endpoint not included."

Answer 2

Well since the domain is the possible values for x(-12,-4,and 8), you would want to substitute these three numbers with x one at a time, and then solve the equation for y every time recording the value of y, all 3 of the y values will represent the range of the function if it's domain is only the three points x=-12, x=-4, and x=8.

so if y=3/4*x+17 then I'll plug in the first value given x=-12 y=3/4*(-12)+17

=(3*-12)/4+17

=-36/4+17

=-9+17

=8(so 8 is part of the range)

now with the second value for x,x=-4 y=3/4*(-4)+17

=(3*-4)/4+17

=-12/4+17

=-3+17

=14(so 14 is the second point in the range)

x=8 y=3/4*8+17

=24/4+17

=6+17

=23(23 is the third and last point in the range)

so the range is (8,14,23)

Let me know if this helped at all :)